In a 12 to 14-inch saute pan, render and brown guanciale until crispy and golden. Do not drain fat from pan; set aside.
Bring 6 quarts of water to a boil and add 2 tablespoons salt. Cook spaghetti according to the package directions, until tender yet al dente, reserving the pasta cooking water.
Reheat the guanciale in the pan with the fat and add approximately 1/4 cup of the pasta cooking water to the pan. Toss in the cooked pasta and heat, shaking the pan, for 1 minute. Add the grated cheese, egg whites, and black pepper and toss until fully incorporated. Divide the pasta among 4 warmed serving bowls. Make a nest in the center for the egg yolk. Gently drop an egg yolk into each serving, season with more freshly ground black pepper and grate additional cheese over the top. Serve immediately.