In a 12- to 14-inch sauté pan, render and cook the guanciale until it is crispy and golden, 6 to 8 minutes. Do not drain the fat from pan and set aside.
Bring 6 quarts of water to a boil and add 2 tablespoons salt. Cook the spaghetti, until tender yet al dente. Drain the spaghetti, 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. Toss in the cooked spaghetti and heat, shaking the pan, until warmed through, about 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.