In a large bowl, add apple juice, 2 tablespoons brown sugar and yeast. Stir well and let sit 10 minutes. Meanwhile, in a medium saucepan, add diced apples and enough water to cover. Over medium heat, cook until apples are crisp tender, about 5 minutes. Remove, strain, transfer to a bowl. Stir in apple liqueur and refrigerate until needed.
After 10 minutes, stir 2 cups flour, oil and salt in the yeast mixture. Using a wooden spoon, continue stirring until it begins leaving the side of the bowl. Loosely cover with a cloth and place in a warm spot to rise for about an hour, or until almost double in bulk. On a well floured work surface, empty dough and knead for 2-3 minutes, or until elastic feeling and it is no longer sticky. Divide into 12 equal pieces. Grease a baking pan with nonstick cooking spray; set aside. Remove cooked apples from refrigerator and stir in apple liqueur; set aside.
Flatten out a dough ball until it is about 3 inches in diameter. Place a tablespoon of apple mixture in the center and pinch until it is sealed. Place, sealed side down, on prepared baking pan. Repeat with remainder of dough and apple mixture, leaving 3 inches between each filled dough ball. Brush the top of each with beaten egg and place in cold oven. Turn heat to 350-degrees F and bake 30-32 minutes, or until lightly browned on top. The filling may ooze out on top, but that is fine. Remove from oven to a plate or rack to cool slightly before serving or cool completely.
* De Kuyper Sour Apple or Apple Pucker Schnapps are especially great in this recipe.