In a large sauté pan, over medium-high heat, melt the butter. Add the sugar, brown sugar, and lemon juice. Stir the mixture for 1 minute, to dissolve the sugars. Add the apples and sauté for 3 minutes. Add the Calvados and carefully flame the apples, shaking the pan back and forth several times. Season the apples with nutmeg, cinnamon, and salt.
Whisk the cornstarch and water together. Add to the apple mixture and bring the liquid to a boil. Reduce the heat to medium and simmer for 2 to 3 minutes, or until the mixture is thick. Cool the mixture completely.
Place 1/2 cup of the filling in the center of each pastry round. Fold the pastry over, forming a half-moon shape. Using a fork or you fingers, crimp the edges of each pastry, sealing the pies completely.
Either heat a large saucepan of oil to 370°F. Deep-fry the pies until golden brown, about 4 to 5 minutes. Stirring occasionally for over-all browning. Remove and drain on paper towels. Sprinkle each pie with the cinnamon sugar.
Or preheat the oven to 375°F. Line 2 baking sheets with parchment paper. Place the pies on the lined sheet pan and sprinkle with the cinnamon sugar. Bake until golden brown, about 15 to 20 minutes. Remove from the oven and serve warm.
Garnish each pie with a scoop of ice cream and a drizzle of caramel sauce.
Yields: 12 servings