Bring a large pot of salted water to a boil. Add the whole wheat fettuccine and cook according to package instructions until al dente, about 10-12 minutes. Drain and set aside.
While the pasta cooks, season the chicken breasts with salt and black pepper on both sides.
Heat olive oil in a large skillet over medium-high heat. Add the chicken breasts and cook for 6-7 minutes per side, or until fully cooked and no longer pink inside. Remove from skillet and let rest for 5 minutes, then slice into strips.
In the same skillet, reduce heat to medium and add the butter. Once melted, add the minced garlic and sauté for 1 minute until fragrant.
Pour in the chicken broth and bring to a simmer. Let it reduce slightly for about 3 minutes.
Lower the heat to medium-low and stir in the Greek yogurt, grated Parmesan cheese, and ground nutmeg. Stir continuously until the sauce is smooth and heated through, about 2-3 minutes. Do not boil to prevent curdling.
Add the cooked fettuccine and sliced chicken to the skillet with the Alfredo sauce. Toss gently to combine and coat the pasta evenly.
Sprinkle chopped fresh parsley over the top and serve immediately.