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 1/4 teaspoon salt and 1/8 teaspoon 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 internal temperature reaches 165°F (74°C). Remove from skillet and let rest for 5 minutes, then slice into thin strips.
In the same skillet, reduce heat to medium. Add the butter and minced garlic. Sauté the garlic for about 1 minute until fragrant, stirring frequently to prevent burning.
Add the chicken broth to the skillet and bring to a simmer. Let it reduce slightly for about 3 minutes.
Lower the heat to medium-low. Stir in the Greek yogurt, grated Parmesan cheese, remaining 1/4 teaspoon salt, 1/8 teaspoon black pepper, 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 sauce. Toss gently to combine and coat the pasta evenly with the sauce.
Remove from heat and sprinkle chopped fresh parsley over the top before serving.