Bring a large pot of salted water to a boil and cook the fettuccine according to package directions until al dente.
While the pasta cooks, season the chicken breast with half of the sea salt and black pepper.
Heat the olive oil in a non-stick skillet over medium-high heat and sear the chicken for 5-6 minutes per side until cooked through, then set aside to rest before slicing.
In the last 3 minutes of the pasta cooking time, add the broccoli florets to the boiling water to blanch them until bright green and tender.
Reserve 2 tablespoons of the starchy pasta water, then drain the pasta and broccoli.
In a small bowl, whisk together the Greek yogurt, grated parmesan cheese, minced garlic, and the reserved pasta water to create a smooth sauce base.
Return the pasta and broccoli to the warm pot over low heat, pour in the yogurt mixture, and toss gently until the sauce coats the noodles evenly.
Slice the chicken and place it on top of the pasta, garnishing with fresh chopped parsley and the remaining salt and pepper.