Cook guanciale with olive oil and 1 teaspoon ground black pepper in a Dutch oven over medium-low heat until almost crisp, about 5 minutes. Reduce heat to low.

Whisk eggs, 3 tablespoons Parmigiano-Reggiano cheese, 3 tablespoons Pecorino Romano cheese, and 1 teaspoon ground black pepper in a bowl. Set aside.

Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water, stirring occasionally until cooked through but firm to the bite, 10 to 12 minutes. Drain, reserving 1 cup of the pasta water.

Pour reserved pasta water and drained spaghetti into the bacon-pepper mixture; stir to combine.

Slowly pour egg mixture into the pasta mixture; cook, stirring constantly until the egg mixture forms a thick sauce, 1 to 2 minutes. Remove from heat.

Stir remaining 3 tablespoons Parmigiano-Reggiano cheese and 3 tablespoons Pecorino Romano cheese into the pasta mixture and serve.