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Preheat oven to 400°F.
Roll out pie dough to a 12-inch circle on a lightly floured surface. Fit into a 10-inch cast-iron skillet, pressing dough up the sides. Refrigerate until ready to use.
Toss together apples, sugar, flour, cinnamon, and nutmeg in a large bowl. Melt butter in skillet over medium heat. Add apple mixture and cook, stirring occasionally, until apples begin to soften, about 10 minutes.
Remove skillet from heat and top with pie dough, tucking edges under and crimping to seal. Cut 4 slits in top of dough. Brush top with egg wash.
Bake in preheated oven until crust is golden brown and filling is bubbly, 30 to 35 minutes. Let cool 30 minutes before serving.