Mix the cooled spaghetti, grated cheeses and eggs together in a large bowl. Sprinkle with salt and pepper. Add the oil to a large nonstick saute pan and heat to medium heat. Once hot, add a handful of the spaghetti mixture to the pan and flatten it out to the edges. Lay the American cheese on top and add another layer of the spaghetti mixture. Cook for about 5 minutes and flip, cooking for another 5 minutes on the other side until golden brown.
Turn out onto a cutting board, slice as you would a pizza and serve.