1. In a large bowl toss together the apples, sugars, lemon juice, cinnamon, salt and nutmeg. Melt the butter in a large skillet over medium heat until it browns. Remove from the heat, add the apple mixture and all juices to the skillet. Stir to coat with the butter and spread in an even layer. Cook, stirring occasionally, until the apples are tender but still hold their shape. The juices should be thick and dark brown, 10 to 13 minutes. Remove from the heat; stir in the vanilla. Cool the filling completely before assembling.

2. Preheat oven to 400 degrees. Line a baking sheet with parchment. In a small bowl whisk together the egg and 1 tablespoon water to make an egg wash. Set aside.

3. Roll the sheet of puff pastry into a 12x14-inch rectangle. Cut in half lengthwise; move one rectangle to the baking sheet. Brush a 1-inch border of egg wash over the perimeter of the dough. Arrange the cooled filling over the “non-egg washed” area. Shingle the apples and spoon some of the sauce over them.

4. Lightly flour the other rectangle and fold in half lengthwise. Cut 1 1/2-inch long slices, 1 inch apart, leaving a 1-inch border on three sides. Lift the folded sheet on top of the filling. Gently unfold it and stretch it over the filling. Press the edges together with your fingers. The very lightly crimp the edges with a fork. Chill for at least 20 minutes.

5. Brush the tart with the egg wash. Sprinkle with the sugar. Bake for 15 minutes and then rotate the baking sheet. Continue to bake until the pastry is puffed and deep golden brown, about 10 to 15 minutes more. Cool for 45 minutes before serving.