Bring a large pot of salted water to a roaring boil and cook the fettuccine according to package instructions until al dente, about 8-10 minutes. Reserve a cup of cooking water before draining.

Season the chicken breasts with salt and black pepper. Heat olive oil in a skillet over medium-high heat. Cook chicken for 6-7 minutes per side until browned and fully cooked, reaching an internal temperature of 165°F. Remove from skillet, let it rest, then slice into bite-sized pieces.

Melt butter in the same skillet. Add sliced mushrooms and sauté for 4-5 minutes until golden brown. Add minced garlic and cook for an additional 1-2 minutes.

Pour in chicken broth to deglaze the pan, scraping up brown bits. Let it simmer for 2-3 minutes. Stir in heavy cream, Parmesan cheese, and dried thyme, simmering for 3-4 minutes until slightly thickened.

Add sliced chicken and cooked pasta to the skillet, tossing to coat in sauce. Use reserved pasta water if the sauce is too thick.

Garnish with freshly chopped parsley if desired and serve immediately while hot.