Special equipment: 2 pastry bags, a 1/2-inch tip and a medium tip
For the puffs: Preheat the oven to 450 degrees F.
Put the butter, sugar, salt and 1 1/2 cups water in a saucepan over medium heat and bring to a simmer, 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 1 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. Smooth the dough peaks with a wet finger.
Bake until puffed, 15 to 20 minutes, then lower the temperature to 350 degrees F and bake until golden, about 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.
For the white chocolate: While the puffs are baking and cooling, bring a small saucepan of water to a simmer. Combine the white chocolate with the heavy cream in a small metal bowl and set it on top of the saucepan without touching the water. Heat, stirring, until the white chocolate melts and the mixture is smooth. Set aside to cool.
For the whipped cream filling: Using a stand mixer or hand mixer and a whisk attachment, beat the heavy cream with the confectioners' sugar until soft peaks form. Whisk in the cooled white chocolate mixture.
To assemble: Add the whipped cream filling to a pastry bag fitted with a medium tip. Fill each puff with a few tablespoons of the mixture through the hole already punched into it. Set aside.
For the caramel: Mix the granulated 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, about 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 (if the caramel hardens, microwave until soft, 45 seconds). Arrange the puffs around the circle on the parchment. 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 2 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.