8 ounces chilled all-butter puff pastry
1/2 tablespoon plus 4 tablespoons plus 3 cups sugar, divided
1 1/4 cups plus 2 tablespoons whole milk, divided
3 1/2 tablespoons unsalted butter
1/2 cup plus 1 tablespoon flour
2 eggs plus 3 egg yolks
seeds from 1 vanilla bean, divided
1 1/2 tablespoons cornstarch
1 1/2 cups well-chilled heavy cream
2 heaping tablespoons confectioner’s sugar
1/2 tablespoon canola oil
Roll out pastry to 1/8-inch thick. Cut out an 8-inch pastry circle; transfer to a parchment paper-lined sheet pan, and prick all over with a fork. Cover and refrigerate.
Heat oven to 425°F. Bring 1/2 tablespoon 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 whole 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-inch 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 the same baking sheet, arranged around the pastry circle, pipe remaining dough into thirteen 1-inch-wide puffs. Press down the pointy tops with a moist fingertip. Bake pastry circle and puffs until beginning to brown, about 12 minutes. Reduce heat to 325°F; bake until dry and golden – 6 minutes for puffs and 10 minutes for pastry. Let the pastry circle and puffs cool.
Heat remaining milk, half the vanilla bean seeds, and 4 tablespoons sugar in a small pot over medium heat until it starts to bubble around the edges. Meanwhile, beat together egg yolks and 2 tablespoons 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. Cook 8 to 10 minutes over medium-low heat until very thick; this is the pastry cream. Place a circle of parchment directly on the surface of the pastry cream and chill.
Beat remaining vanilla bean and heavy cream to soft peaks in a large bowl. Add confectioner’s 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 two 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.
Grease a sheet pan with the canola oil. Bring remaining sugar and ½ cup water to a boil in a small pot over medium heat. Cook, without stirring, until golden brown. 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; allow caramel to drip off the puff, 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 large star tip; pipe cream onto center of pastry in stars. Chill until ready to serve.