Preheat the oven to 400°F.
In a 10-inch oven-safe skillet, combine peaches, cinnamon, nutmeg, and cornstarch. Set aside while you prepare the puff pastry.
Lay the puff pastry sheet onto a clean work surface.
In a small bowl, mix together the egg and water. Brush the egg wash over the pastry*. Cut the sheet into 16 equal-sized squares.
Arrange the puff pastry squares over the peaches in a circular, overlapping pattern, starting at the largest part of the skillet and working your way toward the center. All of the peaches should be as covered as possible.
Bake for 27-30 minutes, or until the pastry has puffed up and browned and the peach filling is bubbly.
Let cool for 5-10 minutes before serving.
Serve with a scoop of ice cream, whipped cream, or even plain.