In a medium saucepan, melt 1 stick of butter over medium heat. Stir in the sugar and ½ cup of brown sugar, until dissolved.

Stir in the sliced apples and cook for about 5 minutes, or until the apples soften, stirring frequently.

In a small bowl, combine the flour and cornstarch. Gradually stir the flour mixture into the apples. Continue cooking for about 3 minutes, or until the mixture thickens slightly, stirring frequently. Remove from the heat.

Roll the pie crust out to your desired thickness and cut into 6-inch circles. Place about 2 tablespoons of the apple mixture in the center of the circles. Lightly brush the outer edge of the crust with the egg. Fold over and crimp the edges to secure. Repeat with the remaining ingredients.

In a medium pot or Dutch oven, pour in enough oil to deep fry the pies and heat to 365 degrees F.

Meanwhile, in a small saucepan, melt the remaining 1 stick of butter over medium heat. Stir in the cream cheese until smooth. Stir in the remaining ¾ cup brown sugar and maple syrup. Cook for about 2 minutes, until the mixture is smooth, stirring constantly.

Place 1 to 2 pies in the oil and fry for a few minutes or until golden brown, flipping as needed. Try to maintain an oil temperature of 350 degrees F. while frying. Strain from the oil and place on a wire rack. Drizzle with caramel sauce and serve warm or at room temperature. Note: you can rewarm the caramel sauce before serving if desired.