Whisk together gluten free flour, xanthan gum, 1/2 teaspoon salt, and 1 tablespoon sugar. Cut in cold butter until mixture resembles coarse crumbs with pea-sized pieces remaining.
Stir vinegar into ice water and drizzle over flour mixture 1 tablespoon at a time, tossing with a fork. Add just until dough holds together when pressed. Divide into two discs, wrap, and refrigerate for at least 45 minutes.
Toss sliced apples with lemon juice. In a large saucepan, combine apples, brown sugar, granulated sugar, cornstarch, cinnamon, nutmeg, and remaining 1/2 teaspoon salt.
Cook apple mixture over medium heat for 5-7 minutes until apples begin to soften and release juices. Spread filling on a baking sheet to cool completely, about 20-30 minutes.
Roll the larger dough disc into a 12-inch circle on a lightly floured surface. Transfer to a 9-inch pie dish, press into bottom and sides without stretching, and trim overhang to 1 inch. Refrigerate.
Roll the second dough disc into an 11-inch circle. Leave whole, cut into strips for lattice, or use cookie cutters for decorative vents. If whole, cut 4-5 slits in the center.
Preheat oven to 425°F. Pour cooled filling into bottom crust, mounding slightly in the center. Dot with butter pieces. Place top crust over filling, trim overhang, and crimp edges together.
Brush top crust with beaten egg and sprinkle with coarse sugar. Place pie on a parchment-lined baking sheet. Bake at 425°F for 20 minutes, then reduce to 350°F for 40-45 more minutes, covering edges with a crust shield if needed.
Pie is done when crust is deep golden brown and filling is bubbling through vents. Cool on a wire rack for at least 2 hours before slicing to allow filling to set.