For the sauce: Whisk the eggs in a large bowl. Fry the bacon in a large frying pan over medium heat until crisp and golden, and then remove and drain on a paper towel. Leave all but 1 tablespoon of the bacon fat in the pan and add the butter. Return the pan to the burner and cook the onions until slightly golden, 2 to 3 minutes. Lower the heat to medium-low and add the half-and-half and cheese. Simmer until the sauce has thickened slightly, about 5 minutes; do not let boil. Let the sauce cool slightly, about 2 minutes. For the pasta: Boil the pasta in salted water until al dente. Slowly pour the cream sauce into the eggs while continuously whisking. Add the hot pasta and toss quickly. The hot pasta will complete the cooking of the eggs. Add the bacon and parsley and toss again. Season with salt and pepper and serve hot.