Preheat the oven to 350 degrees F (175 degrees C). Grease a deep 9x13-inch baking dish with cooking spray.
Place yellow and red potatoes into a large pot and cover with salted water; bring to a boil. Cook until soft, about 10 minutes. Drain.
Combine mozzarella cheese, Cheddar cheese, sour cream, and mayonnaise in a large bowl; mix well. Slowly add potatoes and toss; use a wooden spoon or flat spatula to avoid mashing the potatoes.
Pour potatoes into the prepared pan. Dot butter over the top and cover with foil.
Bake in the preheated oven, 25 to 30 minutes. Remove from oven and uncover. Preheat the oven's broiler. Return to the oven and broil until top is browned, about 5 minutes.
