Cook the fettuccine in a large pot of boiling salted water according to package directions, adding the broccoli florets during the last 3 minutes of cooking.
Reserve 2 tablespoons of the starchy pasta water before draining the noodles and broccoli.
Season the chicken breast with sea salt and black pepper on both sides.
Heat olive oil in a skillet over medium-high heat and sear the chicken until golden and cooked through.
Remove chicken from the pan to rest, then slice it into thin strips.
In the same skillet, reduce heat and sauté the minced garlic for 30 seconds until fragrant.
Whisk together the Greek yogurt and parmesan cheese in a small bowl, adding a splash of the reserved pasta water to thin the sauce.
Add the cooked pasta, broccoli, and yogurt mixture to the skillet, tossing gently over low heat until the sauce is creamy.
Top the pasta with the sliced chicken and fresh parsley before serving.