To make the crumble, combine the flour, sugar, and butter in a stand mixer fitted with the paddle attachment. Mix on low until a uniform dough forms. Roll the dough between two sheets of parchment paper until thin—about ⅛ inch/3 mm. Freeze until firm, then cut into rounds the same size as the piped choux (4 to 5 cm in diameter). Return to the freezer until ready to use.

Preheat the oven to 340°F (170°C). Line a baking sheet with parchment paper.

For the choux pastry, combine the milk, water, butter, sugar, and salt in a medium saucepan. Bring to a boil over medium heat. Remove from heat, add the flour all at once, and stir vigorously until a dough forms. Return to medium heat and stir constantly for 1 to 2 minutes to dry the dough slightly. Transfer to a stand mixer fitted with the paddle attachment and mix on low to cool for a few minutes. Gradually add the eggs, mixing well between each addition, until the dough is smooth, glossy, and holds a soft peak.

Transfer the dough to a piping bag fitted with a small round tip. Pipe 4 to 5 cm rounds onto the prepared baking sheet, spacing them apart. Place a frozen crumble disc on top of each mound of dough. Bake for 30 to 35 minutes, or until puffed and golden brown. Cool completely before filling.

For the vanilla cream, bring the milk to a boil in a medium saucepan. In a separate bowl, whisk together the egg yolks, sugar, cornstarch, and vanilla. Slowly pour the hot milk over the egg mixture while whisking, then return the mixture to the saucepan. Cook over medium-low heat, whisking constantly, until the cream thickens and begins to bubble. Transfer to a bowl, cover with plastic wrap directly on the surface, and refrigerate for at least 2 hours.

To make the caramel, place the sugar in a clean saucepan and add just enough water to moisten it. Cook over medium heat, gently swirling the pan (do not stir) until the caramel turns a deep amber color. Remove from heat.

Working quickly and carefully, dip the tops of the filled puffs into the hot caramel using a paring knife or tongs. Place caramel-side up on a sheet of parchment to cool and harden.

Serve once the caramel has set.