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 large skillet over medium heat and sauté the chicken until golden and cooked through, about 5-7 minutes per side.
Remove the chicken from the skillet and set aside to rest, then slice into thin strips.
In the same skillet, add the minced garlic and sauté for 1 minute until fragrant, then remove from heat.
Steam the broccoli florets until tender-crisp, approximately 4 minutes.
In a small bowl, whisk together the Greek yogurt, grated parmesan, and 2 tablespoons of the hot pasta cooking water until smooth.
Drain the pasta and add it to the skillet with the garlic, then toss in the chicken, broccoli, and the yogurt sauce.
Gently fold everything together over low heat until the sauce is creamy and coats the noodles, seasoning with the remaining salt and pepper.
Garnish with fresh parsley and serve immediately.