Heat a medium-sized iron skillet over medium heat and fry the bacon until completely rendered and crisp, about 15 minutes. Remove from skillet using a slotted spoon and transfer to paper towels.
Cook spaghetti according to box, drain, and return to pot. Add butter and stir until spaghetti is well-coated. Slowly alternate adding the beaten eggs and grated cheeses, stirring constantly, until thick and creamy. Add bacon and pepper, toss to combine, garnish with a generous sprinkle of Parmigiano-Reggiano and serve immediately.