Cook pasta in salted water until al dente; reserve 1/4 cup cooking water, then drain.
Whisk eggs with 1/2 cup Parmesan and a generous pinch of pepper in a bowl; set aside.
Heat oil/butter in a large skillet over medium, sauté garlic 30 seconds, add chicken and peas, warm through.
Add drained pasta to skillet, remove from heat, then slowly pour egg-cheese mixture while tossing constantly to create a creamy sauce.
Add reserved pasta water a little at a time to loosen sauce; adjust salt and pepper, sprinkle remaining Parmesan and serve immediately.