Preheat oven to 350 degrees F (175 degrees C). Spray a 9x13-inch baking dish with cooking spray; place biscuit dough pieces into the prepared baking dish to cover the base of the dish.
Heat a large skillet over medium-high heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes; drain and discard grease, leaving meat in the skillet. Add pasta sauce to beef and stir to combine. Pour sauce mixture over biscuit pieces to cover completely. Sprinkle mozzarella cheese and Cheddar cheese over the top.
Bake in the preheated oven until cheese is bubbling and biscuits are cooked through, 30 to 35 minutes.