3/4 lb. chilled puff pastry
2 3/4 cups plus 6 1/2 tsp. sugar
1 1/4 cups plus 2 tbsp. milk
3 1/2 tbsp. butter
1/4 tsp. salt
1/2 cup plus 1 tbsp. flour
2 eggs plus 3 egg yolks
Seeds from 1 vanilla bean
1 1/2 tbsp. cornstarch
1 1/2 cups cold heavy cream
1/2 tbsp. canola oil
6 tbsp. corn syrup
1. Roll out pastry until 1/8" thick, pressing down hard on the rolling pin (which will prevent the pastry from puffing when baked). Cut out an 8" pastry circle; transfer to a parchment paper–lined baking sheet pan. (Discard scraps.) Cover pastry circle with plastic wrap and refrigerate.
2. Heat oven to 425°. Bring 1/2 tsp. sugar, 1/4 cup milk, butter, salt, and 1/4 cup water to a boil in a small pot. Add 1/2 cup flour and cook, stirring, until it forms into a dough ball, about 20 seconds. Transfer dough to a bowl. Add 2 eggs, one at time, while beating with an electric mixer; continue beating until dough is smooth and holds a soft peak. Transfer dough to a pastry bag fitted with a 3/8" round tip and pipe 3 circles onto the chilled pastry circle: one along the outer edge, one around the middle, and one in the center. On another parchment paper–lined baking sheet, pipe remaining dough into twelve 1"-wide puffs (they'll resemble little peaked caps). Press down the pointy tops with a moist fingertip. Bake pastry circle and puffs until beginning to brown, 12–14 minutes. Reduce heat to 325°; bake until dryish and golden, 6–7 minutes for puffs, 10–12 minutes for pastry. Let the pastry circle and puffs cool.
3. Heat remaining milk, half the vanilla, and 3 1/2 tbsp. sugar in a small pot over medium heat until it starts to bubble around the edges. Meanwhile, beat together egg yolks and 2 tbsp. sugar in a medium bowl until pale and thick; sift in remaining flour and cornstarch and whisk to combine. Slowly drizzle in hot milk mixture while whisking constantly. Strain through a fine sieve into a small pot; cook over medium-low heat until very thick to make a pastry cream, 8–10 minutes. Cover and chill the pastry cream.
4. Beat remaining vanilla beans and heavy cream to soft peaks in a large bowl. Add 1/2 tbsp. sugar; beat to stiff peaks. Transfer 2 cups whipped cream to a bowl, cover, and reserve in the refrigerator. Fold remaining whipped cream into the chilled pastry cream in 2 batches. Transfer the cream mixture to a pastry bag fitted with a very small, round tip. Poke bottom of each pastry puff once with a toothpick to make a small hole; pipe cream mixture into each puff to fill. Pipe the remaining cream onto the pastry, leaving the outer ring exposed, and chill.
5. Grease a sheet pan with the canola oil. Bring remaining sugar, corn syrup, and 3/4 cup plus 2 tbsp. water to a boil in a small pot over medium heat. Cook syrup, without stirring, until golden brown, about 40 minutes. Plunge the pot into cold water to stop the cooking, being careful not to get water in the pot. Working with metal tongs, quickly dip each puff into the caramel to coat all over; then transfer to the sheet pan, puffed side up. Let harden. Reheat caramel, then dip the end of each puff, puffed side down, briefly into the caramel and adhere them to the outer ring of the pastry circle, pressing them down to ensure that they stick. Transfer reserved whipped cream to a pastry bag fitted with a saint-honoré tip; pipe cream onto center of pastry in a herringbone pattern, with the notch of the tip facing up. Serve immediately or chill until ready to serve.
MAKES ONE 8" CAKE