Step 1In a large bowl, toss the apples with the granulated and dark brown sugars. Set aside until the apples start to break down and just begin to release their juices, at least 1 hour and up to overnight.
Step 2In a medium saucepan heated over high heat, melt the butter. As it is melting, use a small paring knife to split the vanilla bean. Scrape the pulp and seeds out of the bean and add to the melting butter, along with the bean pod. Swirl the pan to ensure the butter cooks evenly and doesn't burn. Continue cooking until the bubbles subside and the butter is dark brown with a nutty, toasty aroma. Remove from heat and discard the bean pod.
Step 3Heat the oven to 300 degrees. Cut 12 pieces of kitchen string, each 15 inches long.
Step 4Toss the apples in the bowl with the cornstarch, apple juice, cinnamon, nutmeg and brown butter. Transfer the filling to a rimmed baking sheet, spreading the apples in a single layer. Roast the apples until they're tender but keep their shape, 30 minutes, stirring the apples every 10 minutes for even roasting.
Step 5Transfer the pan to a rack, and cool the filling completely. The filling can be stored, covered and refrigerated, up to 3 days.
Step 6Remove the puff pastry dough from the refrigerator to a lightly floured surface. Cut the dough into circles using a 4 1/2-inch cutter (a small bowl or cake pan will work). You will need 12 circles; save the scraps for another use (stack the scraps and pat down before using; do not press the dough into a ball, or the layers will be lost).
Step 7Using a floured rolling pin, roll each circle to a diameter of 7 inches (the circles do not have to be perfect).
Step 8Scoop one-half cup of the filling and mound it into the center of a circle of dough. Pull up the bottom edges to meet in the center of the filling. Holding the dough with one hand, gather up the edges to meet in the center to pinch it closed like a beggar's purse. Tie a piece of the kitchen string loosely around the purse, giving it about an inch of slack (the dough will puff as it bakes, so the string should not be tied too tight). Repeat to make the remaining borsellini.
Step 9Place the borsellini on parchment-lined baking sheets, spacing them about 3 inches apart. Chill the purses at least 30 minutes, up to 3 days before baking.
Step 10Heat the oven to 400 degrees. Brush the borsellini with egg wash, and sprinkle them liberally with cinnamon sugar.
Step 11Bake the borsellini until the pastry is puffed and golden brown, 45 minutes to an hour, rotating the baking sheets every 30 minutes.
Step 12Remove the sheets to a rack to cool. Dust the borsellini with powdered sugar before serving.