Egg mixture: In a liquid measuring cup or medium mixing bowl, whisk together the eggs, egg yolks, grated cheese, and plenty of freshly-cracked pepper (about 50 turns); set aside. Cook pork: Cook the diced pork in a large sauté pan over medium heat until browned and crispy, about 7-10 minutes (use a pan large enough to eventually hold all of the pasta, preferable non-stick or enameled like this one). Remove the pan from the heat. Use a slotted spoon to transfer the pork to a paper towel lined plate. Don’t clean out the pan - we want the grease for later! If there are more drippings than you feel comfortable with, you can drain some; set aside. Meanwhile, cook the pasta: Cook the pasta: Add 2 quarts water (8 cups) to a large Dutch oven (or stockpot wide enough to fit spaghetti). The water will be very shallow because we want it to be very starchy. Cover and bring to a boil. Uncover and stir in 2 teaspoons salt then add spaghetti. Cook pasta uncovered until 1 minute shy of al dente, stirring often so the spaghetti doesn’t stick. Using tongs, transfer the spaghetti to the pork drippings; set aside. Reserve starchy pasta water.
Temper the eggs: While vigorously whisking the egg mixture with one hand, slowly drizzle 1 cup starchy hot water into the egg mixture a couple tablespoons at a time until incorporated (DON’T STOP WHISKING or the eggs will scramble!). Set aside.
Toss the pasta: Place the pasta over medium heat while tossing with ½ cup starchy pasta water until al dente, about 1 minute, then turn off the heat.
Combine: While continuously tossing the spaghetti, SLOWLY drizzle the egg mixture into the pasta. Continue tossing constantly until the cheese melts and the eggs thicken to form a glossy sauce. Thin sauce with additional pasta water if needed. Add the pork and toss to combine. Immediately transfer to a serving bowl (so the eggs don't continue to cook). Season to taste with freshly cracked salt, pepper and red pepper flakes if desired. Serve immediately.