Bring a large pot of salted water to a boil and cook the fettuccine until al dente according to package instructions.
Reserve 0.25 cup of the starchy pasta water before draining the noodles.
Season the chicken breast on both sides with sea salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat and sear the chicken for 6 to 7 minutes per side until golden and cooked through.
Remove the chicken from the skillet, let it rest for 5 minutes, then slice into thin strips.
In the same skillet, reduce heat to low and sauté the minced garlic for 30 seconds until fragrant.
In a small bowl, whisk together the Greek yogurt, lemon juice, lemon zest, and half of the parmesan cheese.
Add the cooked pasta and the yogurt mixture to the skillet, tossing constantly while adding the reserved pasta water 1 tablespoon at a time until a creamy sauce coats the noodles.
Fold in the sliced chicken, fresh parsley, and fresh basil, then serve immediately topped with the remaining parmesan.