Preparation

Make the filling:

• In a large bowl, toss the apples with the brown sugar, granulated sugar, lemon juice, cinnamon, salt, and nutmeg.
• In a 12-inch skillet, melt the butter over medium heat until the milk solids turn golden brown, 1 to 2 minutes. Remove from the heat, add the vanilla seeds, and stir. Carefully add the apple mixture to the skillet; with a heatproof rubber spatula, scrape all the sugar and spices from the bowl into the skillet. Stir the apples to coat them with the butter and then spread them in a fairly even layer. Return the pan to medium heat and cook, stirring gently with the spatula every few minutes (try not to break the apple slices), until the apples are tender but not mushy (taste one) and still hold their shape, and the juices have cooked down to a fairly thick, brown, bubbling syrup, 10 to 13 minutes. Scrape the apples into a wide shallow dish or onto a baking sheet to cool completely before assembling the jalousie.

Assemble the jalousie:

• Line a large rimmed baking sheet with parchment. In a small bowl, make an egg wash by beating the egg with 1 Tbs. water until well combined.
• Unfold the puff pastry dough on a floured surface, and gently pinch together any seams that have split. With a floured rolling pin, roll the dough into a 12×14-inch rectangle. With a sharp knife, cut the rectangle in half lengthwise to form two 6×14-inch rectangles. Use a long spatula to help you move one of the dough rectangles onto the parchment-lined baking sheet.
  

• Use a pastry brush to brush a 1-inch border of egg wash around the perimeter of the dough. (Save the remaining egg wash.) Arrange the fruit in a 4-inch-wide strip down the length of the dough. (I like to shingle the apple slices in a thick herringbone pattern down the length of the dough; you may need to make a double layer of apples.) Some syrupy apple juices will likely remain in the dish; spoon 2 to 3 Tbs. over the apples. If some of the liquid seeps onto the egg-washed border, don’t worry about it.
  
• Lightly dust the remaining piece of puff pastry with flour and then gently fold it in half lengthwise; don’t crease the fold. Using a sharp knife, cut 1-1/2-inch-long slashes at 1-inch intervals along the folded side of the dough; leave at least a 1-inch border on the remaining three sides. Do not unfold the dough. Using a long spatula, gently lift the folded strip and position it over the fruit-filled dough rectangle, matching up the straight edges.
  

  Slash along the folded side.

• Gently unfold the top piece of dough and stretch it over the filling, matching the straight edges all the way around the perimeter of the dough. Press the edges gently with your fingertips to seal the dough, and then, with a fork, very gently crimp the edges of the dough all the way around the pastry.
  

  Unfold the dough over the filling.

Bake the jalousie:

• Chill the assembled jalousie for 15 to 20 minutes. Meanwhile, position a rack in the lower third of the oven and heat the oven to 400°F.
• Right before baking, brush the top of the jalousie with a very light coating of the remaining egg wash (you won’t need it all) and sprinkle with the demerara, turbinado, or granulated sugar.
• Bake for 15 minutes and then rotate the baking sheet. Continue baking until the pastry is puffed, deep golden brown on top, and light golden brown on the bottom—use a spatula to gently lift the jalousie so you can peek underneath—another 10 to 15 minutes. Immediately transfer the jalousie from the baking sheet to a wire rack to cool for at least 45 minutes. (Instead of trying to move the hot jalousie with a spatula, lift the parchment to move the jalousie to the rack and then carefully slide the paper out from under the pastry.)
• Serve the jalousie slightly warm with crème fraîche, lightly sweetened whipped cream, or vanilla ice cream, if you like.

Make Ahead Tips

The jalousie is best served the day it’s made, but it will keep, wrapped well in aluminum foil, for three days. You can reheat it in a 325°F oven for 5 minutes before serving.