1. Preheat oven to 400 degrees. Line 2 baking sheets with parchment.

2. Make the puffs: Bring milk, water, butter, and salt to a boil in a medium saucepan over medium-high heat. Remove from heat, and whisk in flour. Return to heat, and cook, stirring constantly with a wooden spoon, until mixture pulls away from sides of pan, about 4 minutes. Transfer to the bowl of a mixer fitted with the paddle attachment. Mix on medium speed to cool slightly, about 1 minute. With machine running, add eggs, 1 at a time, beating well after each addition. Transfer dough to a pastry bag fitted with a 1/4-inch round tip.

3. Whisk together yolk and heavy cream. Pipe about one hundred 1-inch puffs (about the size of a quarter) onto each prepared sheet. Gently brush with egg wash. Bake until puffs rise and are golden brown, 20 to 25 minutes. Let cool on sheets on wire racks. (Puffs can be stored at room temperature for up to 1 day.)

4. Transfer caramel cream to a pastry bag fitted with a 1/4-inch round tip. Insert tip of pastry bag into base of each puff, and fill each. Return to sheets in a single layer as you work.

5. Assemble the croquembouche: Dip top half of each filled puff into caramel (be careful not to burn your fingers), letting excess drip back into pan. Transfer to a parchment-lined baking sheet. Let stand until caramel is set.

6. Carefully dip bottom half of 1 puff into caramel, letting excess drip into pan. Transfer puff, hot caramel side down, to a serving platter. Repeat with 9 more puffs, forming a connected ring as you work. Repeat with more puffs, layering rings to form a 6-layer pyramid, using 45 or 50 puffs total. (If the caramel begins to harden, reheat briefly over low heat.)

7. Attach sugar cookies to sides and base of croquembouche, using royal icing or caramel as "glue."

8. To make a second croquembouche, make another batch of caramel, and repeat with remaining filled puffs. (Alternatively, serve the remaining puffs on the side.) Serve immediately, or let stand at room temperature for up to 2 hours.