Combine diced apples, granulated sugar, brown sugar, cinnamon, nutmeg, and lemon juice in a medium saucepan over medium heat. Cook for 5 minutes, stirring occasionally, until the apples soften and release their juices.

Stir together the cornstarch and water in a small bowl, then add the slurry and butter to the pan. Cook for 2 more minutes until the filling thickens and turns glossy. Remove from heat and cool to room temperature.

Unroll the pie crusts on a floured surface and cut into 8 rounds using a 4-inch cutter. Beat the egg with the water for the egg wash. Spoon about 1.5 tbsp filling onto each round, brush the edges with egg wash, fold into a half-moon, and seal firmly with a fork.

Heat the vegetable oil in a heavy pot over medium-high heat to 350°F (175°C). Fry empanadas in batches of 2 to 3 for 3 to 4 minutes per side until deep golden brown. Drain on a paper towel-lined plate for 1 minute.

Sprinkle the cinnamon sugar over the warm empanadas and arrange on a serving plate, golden side up.