Step 1: Cook the Pasta. Bring a large pot of salted water to a rolling boil. Add spaghetti or bucatini and cook according to package directions until al dente. Reserve 1.5 cups of starchy pasta water before draining.
Step 2: Prepare the Eggs and Cheese. In a medium bowl, whisk together 2 whole eggs and 2 egg yolks. Stir in 1/2 cup of the grated Pecorino Romano/Parmesan cheese and 1/2 teaspoon of freshly ground black pepper. Set aside.
Step 3: Crisp the Bacon. In a large skillet over medium heat, cook diced bacon or pancetta until crispy. Remove bacon with a slotted spoon, leaving about 1-2 tablespoons of rendered fat in the pan. Set crispy bacon aside.
Step 4: Sauté the Kimchi. Add the chopped kimchi (and 1-2 tablespoons of its juice) to the skillet with the rendered fat. Sauté for 3-5 minutes, stirring occasionally, until softened and fragrant.
Step 5: Combine Pasta and Kimchi. Drain the al dente pasta (remembering the reserved water!) and immediately add it to the skillet with the sautéed kimchi. Toss well to combine and cool the pasta slightly.
Step 6: Create the Carbonara Sauce. Remove the skillet from heat. Gradually whisk about 1/2 cup of the hot reserved pasta water into the egg and cheese mixture to temper the eggs. Pour the tempered egg mixture over the pasta in the skillet. Toss vigorously and continuously with tongs, adding more pasta water a tablespoon at a time, until a creamy, emulsified sauce forms that coats all the pasta. Stir in the crispy bacon, if using.
Step 7: Season and Serve. Taste and adjust seasoning as needed. Serve the Kimchi Carbonara immediately, garnished with the remaining grated cheese, extra black pepper, and chopped green onions or fresh parsley.