Pulse 2 1/2 cups flour, 2 tablespoons sugar, xanthan gum, and salt together in a food processor; add 3/4 cup butter, 2 eggs, and cold water. Blend until evenly mixed and crumbly but holds together when pinched; add more cold water, 1 tablespoon at a time, if needed. Form dough into a ball, place in a bowl, and chill for 30 minutes.
Preheat oven to 350 degrees F (175 degrees C).
Mix apples, 1/4 cup white sugar, brown sugar, 2 tablespoons flour, lemon juice, vanilla extract, cinnamon, and nutmeg in a large bowl.
Melt 6 tablespoons butter in a saucepan over medium heat until foamy; continue cooking and stirring until butter is brown with a nutty aroma, 3 to 5 minutes.
Pour browned butter over apple mixture; stir to coat.
Roll dough out on a lightly floured surface to 1/8-inch thickness; cut into 4-inch circles.
Whisk 1 egg and 1 tablespoon water together in a bowl; brush egg wash over each circle of dough. Spoon about 2 tablespoons apple filling on 1 half of each circle; fold the other half over the filling. Crimp the edges of each pie together using a fork.
Place pies on a baking sheet; brush tops with egg wash and sprinkle with 1 teaspoon white sugar.
Bake in the preheated oven until pies are lightly browned, about 30 minutes.
