4 1/2 cups, plus 2 tablespoons unbleached all-purpose flour (1 1/4 pounds), plus more for dusting
1/2 cup granulated sugar
1 tablespoon kosher salt
4 sticks (2 cups) unsalted butter, room temperature, cut into tablespoons, plus more for pan
2 large eggs, plus 1 large egg yolk
Filling:
2 sticks (1 cup) unsalted butter, cut into pieces
1 cup light-brown sugar
1 teaspoon ground cinnamon
Pinch of kosher salt
1 large egg, lightly beaten
Directions
Dough: In a small bowl, sprinkle yeast over milk; stir until dissolved. Let sit until foamy, about 5 minutes. Stir in vanilla seeds. In the bowl of a mixer fitted with the dough-hook attachment, mix together 4 1/2 cups flour, granulated sugar, salt, and 1/2 stick butter on low speed until butter is incorporated and mixture resembles coarse meal, 3 to 4 minutes. Add yeast mixture; mix until dough just comes together. Add eggs and yolk; mix until just combined, 2 to 3 minutes (do not overmix).
Turn dough out onto a lightly floured surface, making sure to include any loose bits left at bottom of bowl. Gently knead to form a smooth ball, about 30 seconds. Wrap tightly in plastic; refrigerate at least 2 hours and up to overnight.
In the bowl of a mixer fitted with the paddle attachment, beat together remaining 3 1/2 sticks butter and 2 tablespoons flour. Shape mixture into a 12-by-10-inch rectangle on a piece of plastic wrap. Refrigerate 15 minutes or up to 1 day. On a lightly floured surface, roll out dough to an 18-by-10-inch rectangle, a little over 1/4 inch thick, keeping the corners as square as possible. Remove any excess flour with a dry pastry brush. Remove butter mixture from refrigerator and let stand at room temperature until it is the same consistency as the dough. With a short side facing you, place butter mixture over two-thirds of the dough. Fold the unbuttered third over as you would a business letter, followed by the remaining third. This seals in the butter.
Roll out dough again to an 18-by-10-inch rectangle, then fold into thirds as described above; refrigerate 1 hour. This completes the first of three turns. Repeat rolling and folding process two more times, refrigerating at least 1 hour between turns. To help you remember how many turns have been completed, mark the dough after each one: make one mark for the first turn, two for the second, and three for the third.
Wrap tightly in plastic and refrigerate at least 4 hours and up to overnight. Dough can also be frozen, tightly wrapped in plastic, up to 2 weeks (thaw in refrigerator overnight before using).
Filling: Melt butter in a small saucepan over medium heat. Continue to cook, stirring constantly, until deep golden brown, about 11 minutes. Immediately transfer to a large heatproof bowl and let cool to room temperature, about 10 minutes. Stir in brown sugar, cinnamon, and salt.
Butter a 14-inch round cake pan. On a lightly floured surface, roll out dough to a 20-by-16-inch rectangle. Spread filling on lower half of dough, leaving a 1/2-inch border. Fold top half of dough over filling, pressing edges to seal. Using a pizza wheel or sharp knife, cut dough lengthwise into six equal strips. Coil strips into S shapes and arrange in a circle in pan. Loosely cover with plastic wrap and set aside to rise in warm spot until puffy and full of air, about 45 minutes to 1 hour.
Preheat oven to 375 degrees. Brush top and edges of dough with egg wash. Bake until golden brown, firm to the touch, and an instant-read thermometer inserted into center reads 205 degrees, 35 to 40 minutes. Let cool in pan 5 minutes. Run a knife around edges of pan to loosen, then invert onto a wire rack. Re-invert onto a second rack so top side is up; let cool completely. Pastry is best served the same day.
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