Whisk eggs, yolks, Pecorino, Parmesan and 1/4 teaspoon pepper in a bowl. Cook spaghetti in water with the tablespoon salt until just tender, reserving 1 cup hot pasta water before draining.
Meanwhile, cook pancetta in a large skillet over medium heat for 5 to 6 minutes until browned. Transfer to a plate, leaving about 1 tablespoon drippings in the skillet.
Season chicken with its salt and remaining pepper and cook in the drippings for 5 to 7 minutes until at least 165°F. Remove the skillet from heat and add hot spaghetti and pancetta.
Whisk 1/4 cup hot pasta water gradually into the egg mixture, then pour over the pasta and toss continuously. Return to very low heat only as needed, stirring and adding more water by the tablespoon until the sauce is silky and reaches 160°F without boiling.
Divide among warm bowls immediately so the egg sauce remains smooth around the chicken and spaghetti.