Season both sides of the chicken breasts with salt and black pepper.
Heat olive oil and 1 tablespoon of butter 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 the skillet and set aside.
Reduce heat to medium and add the remaining 2 tablespoons of butter to the skillet.
Add the sliced mushrooms and cook for 5 minutes, stirring occasionally, until they release their moisture and begin to brown.
Add the minced garlic to the mushrooms and cook for 1 minute until fragrant.
Pour in the chicken broth, scraping the bottom of the skillet to loosen any browned bits.
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 cooked chicken breasts to the skillet, spoon some sauce over the top, and cook for an additional 2 minutes to heat through.
Sprinkle chopped fresh parsley over the dish before serving.