Preheat oven to 375°F (190°C). Line a baking sheet with parchment paper.
In a medium saucepan, toss diced apples with lemon juice, brown sugar, cinnamon, and nutmeg. Cook over medium heat for about 5 minutes, stirring often.
In a small bowl, mix the cornstarch and water until smooth. Stir into the apple mixture. Continue cooking for another 2–3 minutes, until thickened. Let cool.
Unroll the pie dough on a lightly floured surface. Cut out circles using a large cookie cutter or the rim of a glass.
Spoon a small amount of the apple filling onto the center of half of the circles. Top each with another dough circle and press edges with a fork to seal.
Brush the tops with the beaten egg and sprinkle with granulated sugar.
Bake for 18–22 minutes, or until golden brown. Cool slightly before serving.