Pat the chicken breasts dry with paper towels. Season both sides with salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken breasts 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. Add the butter to the skillet. Once melted, add the minced garlic and sauté for 30 seconds until fragrant.
Add the sliced mushrooms to the skillet and cook for 5-7 minutes, stirring occasionally, until they are browned and softened.
Pour in the chicken broth and use a wooden spoon to scrape any browned bits from the bottom of the skillet. Let it simmer for 2 minutes.
Stir in the heavy cream, grated Parmesan cheese, dried thyme, and crushed red pepper flakes. Cook for 3-4 minutes, stirring frequently, until the sauce thickens slightly.
Return the cooked chicken breasts to the skillet. Spoon some sauce over the chicken and cook for an additional 2 minutes to heat through.
Sprinkle chopped fresh parsley over the top before serving.