Preheat oven to 400°F. Line a rimmed baking sheet with parchment paper.
On a lightly floured counter, roll puff pastry into a roughly 11- × 16-inch rectangle approximately 1/8 inch thick. Transfer to the baking sheet and place in the freezer while you prepare the filling.
In a medium bowl, cream together the softened butter and 1/3 cup of the sugar. Add 1 egg and beat until smooth. Stir in the ground almonds and flour until a smooth paste forms. This frangipane may be made up to 3 days in advance and stored in the refrigerator.
Peel and core the apples, then cut into quarters and slice into 1/4-inch wedges. Tumble the apples into a medium bowl and toss with another 1/3 cup sugar and the nutmeg.
Spread the almond frangipane over the puff pastry, leaving at least a 3-inch border on all sides. Arrange the apple slices over the frangipane, overlapping and tucking them in close together.
Fold the sides of the puff pastry up and over the apples in an easy, free-form way, making an edge of at least 2 inches. The beauty of a galette is that it doesn’t have to look perfect. Pinch the pastry folds firmly together to seal them.
Brush the apples with the melted butter. Lightly beat the remaining egg and brush it over the puff pastry. Sprinkle everything with the remaining tablespoon sugar. Bake for 50 to 55 minutes, until the pasty is golden and the apples are soft when poked with a fork. Cool slightly, then sprinkle with almonds and serve warm. The galette is best enjoyed on the same day it is made.