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 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 are browned and softened.
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. Let it simmer for 3 minutes.
Stir in the heavy cream, grated Parmesan cheese, dried thyme, and crushed red pepper flakes. Cook for 3-4 minutes until the sauce thickens slightly.
Return the cooked chicken breasts to the skillet, spooning some sauce over the top.
Simmer for 2-3 minutes to heat the chicken through and meld the flavors.
Sprinkle chopped fresh parsley over the dish before serving.