PREPARE THE BISCUIT: Preheat oven to 420 degrees F. Place the whole eggs and yolks in a medium-sized mixing bowl. Add 1/3 cup of the granulated sugar and the milk. Whip with an electric mixer on medium-high speed, until the mixture lightens in color and triples in volume, about 6 minutes. Place the egg whites in a large mixing bowl, and whip at medium speed until foamy. Make French meringue by adding the remaining 2 tablespoons granulated sugar, 1 tablespoon at a time. When all the sugar has been added, increase mixer speed to medium-high and whip to stiff but not dry peaks, about 5 minutes.
Use a rubber spatula to fold the egg mixture in batches into the meringue, incorporating it gradually so as not to deflate the meringue. When all the yolk mixture has been incorporated, place the flour in a fine-mesh sieve and sift over the mixture. Gently fold this in with the rubber spatula. Spread the batter evenly on a parchment-covered baking sheet, liberally dust the top of the biscuit with powdered sugar and place in the oven. Bake just until biscuit begins to brown on top, about 5 minutes. This will lock in the moisture, allowing you to roll the cake later without cracking it. Immediately after removing from the oven, run a knife around the inside of the baking sheet to loosen the cake from the sides. Remove cake from baking sheet and allow it to cool.
PREPARE CHOCOLATE PASTRY CREAM: Make pastry cream as directed and let cool. Melt the chocolate in a double boiler or a mixing bowl placed over a bowl of simmering water, stirring it occasionally as it melts. When all of the chocolate is melted and smooth, remove it from heat, and add the cold pastry cream to the warm chocolate. Mix them together with a rubber spatula.
PREPARE THE ROULADE: Place the biscuit on your work surface and soak with the flavored syrup. Spread the chocolate pastry cream evenly over the cake, spreading it all the way to the edges. Start at one long end and, rolling toward you, roll it into a tight cylinder. Place the roulade on a parchment covered baking sheet and place in the refrigerator for a few hours to set.
PREPARE THE MUSHROOMS: Preheat oven to 250 degrees. Whip the egg whites on medium speed until foamy. Add the granulated sugar 1 tablespoon at a time, until you have volume, then add the rest of the sugar. Increase mixer speed to medium-high and whip until you have stiff but not dry peaks. Gently fold in the powdered sugar. Place the meringue in a large pastry bag fitted with a 1/4-inch plain tip. Pipe the mushroom pieces onto a parchment covered baking sheet. Make the stems by squeezing out a dime-sized mound of meringue, then pull the pastry bag straight up, leaving a small tail. Make the caps by piping 1/2-inch mounds onto the sheet. Sprinkle the caps with a little cocoa powder and place in the oven. Bake until firm and dry, about 1 hour, leaving the oven door cracked so moisture can escape. Remove from the oven and cool on a rack.
TO FINISH THE BUCHE:
PREPARE CHOCOLATE TREES: Make a cornet from parchment paper and fill half full with tempered chocolate. Pipe a long thick chocolate raindrop on a sheet of parchment paper. Draw a cake comb through the chocolate, starting at the center of the raindrop and moving from left to right. Repeat, starting at the center and moving from right to left. You should have a chocolate pine tree. Repeat for as many trees as you want. Allow the chocolate to set before peeling the parchment paper from it.
PREPARE THE CHOCOLATE FENCE: Fill a cornet half full with the tempered chocolate, and pipe a 3 to 4-inch long railroad track, with the outside rails spaced about 2 inches apart and the inside rails spaced about 1/2-inch apart onto a sheet of parchment paper. Allow the chocolate to set before peeling the parchment paper from it.
PREPARE THE COFFEE BUTTERCREAM: Place the egg yolks and whole egg in a large mixing bowl, and whip until thick, light, and tripled in volume, 5 to 7 minutes. Pour the water and granulated sugar into a 1-quart heavy-bottomed saucepan and place over medium heat. Insert a candy thermometer into the mixture. The sugar is ready when it reaches 250 degrees, or the soft ball stage. Pour the cooked sugar down the side of the bowl into the whipping eggs. Continue whipping the mixture until the outside of the bowl is warm but not hot, about 2 or 3 minutes. Add the butter all at once, and beat on medium speed until incorporated. Now whip on medium-high speed until thick, smooth and shiny, about 10 minutes. Remove about 1/3 cup and set aside. Add the coffee extract to the remaining buttercream and whip until well combined.
ASSEMBLE THE BUCHE: Place the rolled cake on the platter you will use to serve it. Use a sharp knife to slice a 11/2-inch-thick slice from each end of the roll. These will become gnarls on the log. Spread a thick layer of buttercream over the buche. Place the ends or gnarls on top of the log, 1 at each end and cover with buttercream. Run a cake comb through the buttercream along the length of the buche, to make it look like bark. Place the buche in the refrigerator for about 15 minutes to let the buttercream set.
MAKE THE IVY AND FLOWERS: Divide the unflavored buttercream evenly among 3 bowls and color each with 1 of the food colorings. Make 3 cornets from triangles of parchment paper, and fill one with pink buttercream, one with green, and one with yellow. Remove the buche from the refrigerator. Use the green icing cornet to make the ivy, by piping long curved lines along the length of the buche.
Place small yellow dots along this stem, for the flower centers. Surround these with pink dots for the flower petals. Cut the tip of the cornet diagonally to make a point and fill it with green icing. Squeeze a dab of green buttercream out of the corner of each leaf.
FINISHING THE BUCHE: Stick together the caps and ends of the mushrooms, using a dab of buttercream. Place the mushrooms, chocolate trees and chocolate fence on the buche. Dust the buche with powdered sugar and sprinkle with chocolate shavings. Refrigerate for 1 hour,then serve and enjoy!
Sift together the cornstarch and half the sugar in a medium bowl, add the egg yolks and whisk until well combined. Pour the milk and remaining sugar into a 2-quart nonreactive heavy-bottomed saucepan and place over medium-high heat. While milk is heating, slice the vanilla bean in half lengthwise. Separate the seeds, and add both seeds and skin to the milk. Bring it to a boil. Carefully pour about half of the milk into the egg mixture, whisking so eggs don't scramble. Pour this mixture back into the saucepan and continue to whisk, as pastry cream thickens. Bring to a boil and continue to whisk and cook for 2 minutes more. Remove from heat, strain through a fine-mesh sieve, and add butter in small chunks if desired. Stir butter into cream until well incorporated. Pour pastry cream into an airtight container and put plastic wrap on top to prevent a skin from forming. Store in refrigerator for up to 3 days, until ready to use.
Place the water and sugar in a heavy saucepan over medium-high heat and bring to a boil, stirring occasionally. The sugar should completely dissolve. Remove from heat and pour into a mixing bowl. Cool before adding the liqueur.
TEMPERED CHOCOLATE: When you melt chocolate, the molecules of fat separate. To put them back together, you temper it. One way to temper is to place the chocolate in the microwave, and microwave it on high power for 30 seconds at a time until the chocolate is melted. As it recrystallizes, it becomes tempered.