Bring a large pot of salted water to a boil and cook fettuccine according to package directions (8-12 minutes). During the final 3 minutes, add broccoli florets to the pasta water. Drain together and set aside.

While pasta cooks, heat olive oil in a large skillet over medium-high heat. Add chicken pieces in a single layer without overcrowding. Cook undisturbed for 3-4 minutes until golden. Flip and cook another 3-4 minutes until internal temperature reaches 165°F. Add minced garlic during the last minute of cooking.

In a mixing bowl, whisk cornstarch with 2 tablespoons of milk until completely dissolved. Add remaining milk, Greek yogurt, Parmesan cheese, salt, pepper, and nutmeg, whisking until completely smooth with no lumps.

Reduce heat to medium and pour sauce mixture into skillet with chicken. Stir constantly, scraping bottom to prevent sticking. Cook 3-5 minutes until sauce thickens and coats the back of a spoon. Add milk one tablespoon at a time if sauce becomes too thick.

Add drained pasta and broccoli to skillet and toss until everything is evenly coated with sauce. Let rest for 2-3 minutes, then serve immediately, garnished with extra Parmesan and fresh cracked black pepper if desired.