Place potatoes and bouillon cube in a large pot and cover with water. Bring to a boil, then reduce heat to medium-low and simmer until tender, 10 to 15 minutes.
Drain and return potatoes to empty pot.
Whisk together cream of chicken soup, milk, garlic powder, onion powder, salt, and black pepper in a bowl.
Pour soup mixture over potatoes and mix.
Pour potato mixture into prepared casserole dish.
Stir provolone cheese and Parmesan cheese into potato mixture. Top with a layer of Cheddar cheese.
Bake in preheated oven until bubbling and cheese topping is lightly browned, about 25 minutes.
