Whisk eggs, yolks, cheeses, and 1/4 teaspoon pepper. Cook spaghetti with tablespoon salt until just tender; reserve 1 cup hot water and drain.
Meanwhile, brown pancetta in large skillet over medium 5 to 6 minutes. Plate; leave about 1 tablespoon drippings.
Season chicken with its salt and remaining pepper; cook in drippings 5 to 7 minutes to at least 165°F. Remove skillet from heat; add hot pasta and pancetta.
Gradually whisk 1/4 cup hot water into egg mixture. Pour over pasta; toss continuously. Use very low heat only as needed, stirring and adding water by tablespoon until silky and 160°F without boiling.
Immediately divide among warm bowls to keep egg sauce smooth around chicken and pasta.