For the Cream Filling: Prepare the cream fillings: Sprinkle the gelatin over 1/4 cup cold water in a bowl and set aside to bloom.
Place the milk in a pan. Halve the vanilla beans lengthwise; scrape out the seeds with a paring knife, then add the seeds and pods to the milk. Bring to a simmer, then cover and remove from the heat.
Whisk the egg yolks, sugar, cornstarch and salt in a medium bowl until smooth.
Remove the vanilla pods from the milk. Gradually whisk one-third of the warm milk into the egg mixture.
Whisk the egg mixture into the remaining milk in the pan. Cook over medium heat, whisking constantly, until the mixture boils and thickens, 6 minutes. Continue boiling until the mixture is custard-like, 3 minutes.
Remove the pan from the heat. Slice the butter into pieces and whisk into the custard, then stir in the gelatin mixture.
Microwave the chocolate until melted, 2 to 3 minutes, stirring. Mix 2 tablespoons hot water with the espresso. Transfer half of the cream filling to a bowl, then stir in the chocolate and espresso.
Transfer the remaining cream filling to another bowl. Press plastic wrap directly onto the surface of the chocolate and vanilla creams and refrigerate until cold and firm, at least 2 hours.
When ready to fill the puffs, whip the heavy cream to soft peaks with a mixer. Fold half into the vanilla cream filling and half into the chocolate filling.
Transfer each filling to a large pastry bag with a 1/4-inch tip and set aside. (To do this cleanly, cuff the open end of the pastry bag over your hand.)
For the Pastry Puffs: Unfilled puffs can be frozen for up to a week. To re-crisp, thaw, then brush with a beaten egg; bake 5 minutes at 350 degrees.
While the fillings chill, make the pastry puffs: Preheat the oven to 450 degrees. Bring 1 1/2 cups water, the butter, sugar and salt to a simmer in a saucepan over medium heat, stirring to melt the butter.
Remove from the heat and stir in the flour with a wooden spoon to make a paste. Return to the heat and cook, stirring constantly, until the paste is shiny and pulls away from the pan, 6 to 7 minutes. Cool slightly.
Transfer the paste to a stand mixer and beat with the paddle attachment on medium-low speed to cool, 1 minute. Beat in the eggs one at a time. Scrape down the sides of the bowl as needed.
Transfer the dough to a large pastry bag with a 1/2-inch tip. Line 2 baking sheets with parchment paper, dabbing a bit of dough under the corners to keep the paper in place. Pipe 1 1/2-inch balls of dough onto the paper (about 48 total).
Smooth the dough peaks with a wet finger. Bake until puffed, 15 to 20 minutes, then lower the temperature to 350 degrees and bake until golden, 15 minutes. Turn off the oven; keep the puffs inside 10 minutes to dry out. Pierce each puff with a dry piping tip; transfer to a rack to cool.
To Assemble: Fill half of the puffs with chocolate cream and half with vanilla: Insert the tip of the pastry bag into the hole and squeeze until full. Chill the filled puffs 30 minutes before assembling the tower.
Make the caramel: Mix the sugar, corn syrup and 1 cup water in a saucepan, cover and bring to a boil over high heat; don't stir. Uncover and boil, swirling the pan, until the syrup turns deep amber, 20 minutes.
Immediately dip the bottom of the saucepan in a large bowl of ice water for a few seconds to stop the cooking.
Transfer the caramel to a liquid measuring cup and cool slightly (it should still be liquid). Be careful-the caramel will still be hot!
Draw a 7-inch circle on parchment paper. Partially dip each filled puff into the caramel and let the excess drip off. Arrange the puffs around the circle. If the caramel hardens, microwave until soft, 45 seconds.
Fill the base circle with more puffs for stability, then continue building a conical tower of smaller circles. Top the tower with a single puff. Line your work surface with parchment paper-the next step can be messy.
Dip the tip of a fork into the caramel and quickly wave it in circles around the tower to create a web of caramel strands. Repeat. Let set, then slide two spatulas under the paper and transfer the croquembouche to a platter. Tear off the excess paper around the base.
To serve, crack the caramel web with the back of a knife, then dismantle the tower and transfer the cream puffs to plates.