Preheat oven to 350 degrees F (175 degrees C). Grease a 9x13-inch baking dish.
Break off 2-inch pieces of dough and flatten each piece as thin as possible without breaking using a rolling pin. Spoon about 1 teaspoon turkey, 1 teaspoon stuffing, 1 teaspoon mashed potatoes, and 1 teaspoon vegetables onto each piece of flattened dough. Gently close dough around filling, sealing edges together and forming a ball. Arrange filled dough balls in the prepared baking dish; sprinkle with Cheddar cheese.
Bake in the preheated oven until dough balls are cooked through, 30 to 40 minutes. Invert monkey bread onto a serving dish.