Season both sides of the chicken breasts with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat.
Add the chicken breasts to the skillet and cook for 5-6 minutes on each side until golden brown and cooked through. Remove chicken from skillet and set aside.
Reduce heat to medium and add the butter to the same skillet.
Once the butter has melted, add the sliced mushrooms and cook for 5 minutes until they are browned and tender.
Add the minced garlic to the mushrooms and cook for 1 minute until fragrant.
Pour in the chicken broth, scraping up any browned bits from the bottom of the skillet.
Stir in the heavy cream, Parmesan cheese, dried thyme, and crushed red pepper flakes.
Simmer the sauce for 3-4 minutes until it thickens slightly.
Return the chicken breasts to the skillet and spoon the sauce over them.
Cook for an additional 2-3 minutes to heat the chicken through and allow flavors to meld.
Sprinkle chopped fresh parsley over the top before serving.