Bring 6 quarts of water to boil in a large pot, and add 2 tablespoons salts.
Meanwhile, combine the olive oil and guanciale in a 12- to 14-inch sauté pan set over medium heat, and cook unti the guanciale has rendered its fat and is crispy and golden. Remove from the heat and set aside (do not drain the fat).
Cook the spaghetti in the boiling water until just al dente. Scoop out ¼ cup of the pasta cooking water and set aside. Drain the pasta.
Add the reserved pasta water to the pan with the guanciale, then toss in the pasta and heat, shaking the pan, for 1 minute. Remove from the heat, add 1 cup of the Parmigiano, the egg whites, and pepper to taste, and toss until thoroughly mixed.
Divide the pasta among four warmed serving bowls. Make a nest in the center of each one, and gently drop an egg yolk into each nest. Season the egg yolks with more pepper and sprinkle the remaining ¼ cup Parmigiano over the top. Serve immediately.
Combine the sugar, salt, peppercorns, and thyme leaves in a small bowl. Put the hog jowls in a nonreactive casserole and coat with the mixture, rubbing gently. Cover and refrigerate for 5 to 7 days.
Remove the cheeks from the casserole and hang them, using butcher’s twine, in the refrigerator for at least 3 weeks. The cheeks should be firm and dry, with a slight give. Slice and use like bacon or pancetta; refrigerated, it keeps for weeks.