In a pot over high heat, boil salted water. Cook pasta according to package directions; drain and set aside, reserving ¼ cup pasta water.
In a pan over medium-high heat, heat butter and brown garlic, stirring constantly, 2 minutes. Add chicken to pan and brown, turning occasionally, 5 to 6 minutes.
In a small bowl, whisk together eggs, yolk, Parmesan, and a pinch each of salt and freshly ground black pepper. To the chicken, add pasta and pasta water; turn off heat and add egg mixture and parsley, stirring until egg is thick, but not completely cooked through, 2 to 3 minutes. Serve with additional Parmesan and a drizzle of olive oil, if desired.