Bring a 12-quart pot of heavily salted water to a boil over high heat. Cook spaghetti in the boiling water, stirring occasionally, until tender yet firm to the bite, about 12 minutes. Drain pasta, reserving 2 cups of cooking water.
Return pasta and cooking water to the pot over low heat. Add 3 1/2 cups Pecorino Romano cheese, butter, and pepper. Stir until cheese has melted and formed a creamy sauce, about 5 minutes. Season with salt.
Serve with additional cheese, if desired.
