Heat 6 qt. water in a large pot over high. When water starts to steam add 3 tbsp. salt and cover pot with a lid (this will bring water to a boil faster)
While you are waiting for the water to build, cut 4 oz. guanciale into 1" x ¼" strips, and grate 2 oz parmesan cheese and set aside one-quarter of cheese for later.
Whisk 4 eggs yolks and 2 whole eggs in a medium bowl until no streams remain then stir in remaining cheese. Add several cranks of pepper and set aside.
Heat 2 Tbsp. oil in a large Dutch oven or other heavy pot over medium. Add guanciale and cook, stirring occasionally, until crips around the edges, 7-10 minutes.
Remove pot from heat. Using a wooden spoon fish out guanciale and transfer into a small bowl. Pour fat into a heatproof measuring cup, then add back about 3 Tbsp. to pot. Discard remaining fat.
Cook 1 lb. pasta in boiling water, stirring occasionally, 2 minutes shy of package instructions. Just before pasta is finished, scoop out 1¾ cups pasta cooking liquid with the same heatproof measuring cup.
Add 1 cup reserved pasta cooking liquid to Dutch oven and bring to boil over medium-high. Drain pasta into a colander, then transfer to Dutch oven.
Cook pasta, stirring constantly and vigorously until al dente and water is reduced by about half, about 2 minutes. Remove pot from heat.
Remove ¼ cup pasta cooking liquid into reserved egg mixture, then very slowly stream into dutch oven, stirring constantly, until cheese is melted and egg is thickened to form a glossy sauce. Season with salt if needed. Thin sauce with remaining ½ cup pasta cooking liquid, adding a tablespoon at a time until it’s the consistency of heavy cream.
Mix guanciale and serve. Top with pepper and reserved cheese.