Preheat the oven to 400 degrees F.
Cut the sweet potato into rough 1 1/2-inch cubes and the red potato slightly smaller (1-inch cubes) as the sweet potato will cook more quickly. Halve the red onion, then cut each half into 4 to 6 segments, discarding any tough outer skin. Seed the peppers and cut into 1-inch squares, and separate the cloves of garlic. Put everything into a 2-quart flameproof casserole dish, or whatever you want to use (it should be big, otherwise use 2 dishes) and, using your hands, give the vegetables a good coating of olive oil. Season with black pepper, but no salt as the cheese will make it salty (and anyway, the salt will make water leach out).
Bake for 45 minutes, by which time the vegetables should be cooked through and here and there tinged with brown. You'll need to turn the oven up to maximum heat or light the broiler for the endgame: so place the thinly sliced cheese on top of the bake, and put it back in the very hot oven or under the broiler until the cheese has melted and turned slightly brown on top, about 5 to 10 minutes. Serve straight out of the casserole dish.