PEAR-AND-FRANGIPANE PASTRIES

Steps:

• 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, 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).
• Poached Pears:Combine wine, lemon juice and zest, sugar, cinnamon, vanilla seeds, and 2 cups water in a large saucepan. Bring to a boil; cook 5 minutes. Add pears, reduce to a simmer, and cover with a parchment round to keep fruit submerged; cook, occasionally turning pears by gently rotating the stems with your fingertips, until pears are tender, about 20 minutes. Transfer pears to a bowl. Return poaching liquid to a boil and continue boiling until syrupy and reduced by half, 10 to 12 minutes. Pour syrup over pears and refrigerate, covered, at least 4 hours and preferably overnight.
• Frangipane:In the bowl of a mixer fitted with the paddle attachment or a mini food processor, beat together butter and sugar until light and fluffy. Add egg, almonds, rum, almond extract, and flour; beat until smooth.
• Using a slotted spoon, remove pears from poaching liquid and pat dry with paper towels. On a lightly floured surface, roll out dough to a 20-by-12-inch rectangle, about 1/4 inch thick. Using a 3 1/2-inch round cutter, cut out 16 rounds and divide between two parchment-lined baking sheets. Fill each pear half with about 2 teaspoons frangipane and place cut-side down in centers of rounds. Cover with plastic wrap and let stand in a warm place until doubled in bulk, about 45 minutes.
• Preheat oven to 375 degrees. Brush rounds with egg wash, carefully avoiding pears. Bake, rotating halfway through, until pastries are evenly browned, 20 to 25 minutes. Transfer to a wire rack and let cool completely. Serve at room temperature.