For the inlay paste: In the bowl of an electric mixer, cream together the butter and confectioners' sugar. Add the egg whites and beat again until incorporated, about 2 minutes. Sift the flour into the mixing bowl and beat on low speed until just combined. Divide the mixture between four bowls. Tint the first bowl of paste with the juniper green food color; begin with 1/8 teaspoon, mixing and adding a little color at a time, until a dark green color is achieved. Tint the second bowl of paste with a dab (the amount picked up on the end of a toothpick) of juniper green food color and dabs of neon yellow food color. Mix until a light avocado color is achieved. Tint the third bowl with two dabs of neon yellow gel food color and mix until bright yellow is achieved. Tint the fourth bowl using the brown food color; begin with 1/8 teaspoon and add more until a dark brown color is achieved. Transfer each color to a piping bag with a small hole snipped in the end. You may also fit each bag with a small round decorator tip, if desired.
Cut a sheet of parchment paper to fit inside a 15-by-10-inch jelly roll pan. Print the Avocado Emoji Template (link below) and trace the pattern onto the piece of parchment paper using a pencil. Flip the paper over so that the side with the drawn emojis is touching the pan. You should be able to see the images through the paper. Pipe a line of yellow paste around the face (oval part) of the avocado, then use the brown paste to fill in the round shape in the center (the avocado pit). Fill in the yellow outline using the light green paste. Finally, use the dark green to pipe the avocado shell-shape. Transfer the pan with the piped images to the freezer while you prepare the cake batter.
For the cake roll: Preheat the oven to 350 degrees F.
In the bowl of an electric mixer, beat the eggs on high speed using the whip attachment until they lighten in color considerably and have the appearance of yellow cake batter, about 5 minutes. With the mixer still running, slowly add the granulated sugar and oil to the whipped eggs. Next, add the buttermilk and vanilla extract.
Sift together the flour, baking powder and salt. Slowly add the dry ingredients to the liquid ingredients. Mix until well combined. Pour the batter into the prepared jelly roll pan and over the inlay paste emojis. Smooth the batter using a rubber spatula to distribute it evenly.
Lay a large cotton tea towel on a work surface. Sprinkle the towel with confectioners' sugar and rub the sugar into the towel with your hands. Set aside.
Bake until the cake springs back when pressed with fingers, 10 to 12 minutes (check at 10 minutes). Immediately turn the cake out onto the tea towel--do this without delay! Peel off the parchment to reveal the avocado design. Gently flip the cake over so that the avocados are against the tea towel. Immediately roll the cake into the tea towel, beginning at the narrow end. Roll as evenly as possible. Place the cake in the towel on a refrigerator shelf and chill for 1 hour.
For the avocado frosting: Halve and pit the avocados. Using a spoon, scoop the avocado from the skins and place in the bowl of an electric mixer. Squeeze the juice from the lemon half into the bowl over the avocados. Mix on high speed until the avocados are creamed with just a few pea-sized solid pieces of avocado in the mixture. Add the confectioners' sugar a little at a time until thick, spreadable frosting forms.
For the cake assembly: Remove the cake from the refrigerator and gently unroll; allow the most tightly rolled end to remain slightly curled. Apply the avocado frosting to the cake in a smooth even layer using an offset spatula. Re-roll the cake and then roll it up in the tea towel. Refrigerate until firm, about 1 hour.
Combine the granulated sugar and 1/4 cup water in a saucepan and place over medium-high heat. Heat, stirring, until the sugar is melted. Transfer to a small bowl and allow to cool to just warm.
Remove the cake from the refrigerator. Brush the outside of the cake with the simple syrup using a pastry brush. Apply the syrup in an even layer to remove the dusty confectioners' sugar appearance. Serve the cake immediately or cover with plastic wrap and refrigerate until ready to serve.