Cook the spaghetti according to package directions. Do not rinse.
While the pasta is cooking brown the butter by placing into a saucepan over medium heat. As the butter melts, it will begin to foam. Remove the foam with a spoon. Cook the butter until it browns.
To prepare the pasta, pour pasta into a bowl. Pour the browned butter over the pasta, and then add the two kinds of cheese. Toss pasta until it is well coated. Serve immediately. This dish does not reheat well.