Prepare the Chicken: Season the chicken breasts generously with salt and pepper on both sides.
Cook Chicken: In a large skillet, heat olive oil and butter over medium-high heat. Add the chicken breasts and cook for about 5-6 minutes per side, or until golden brown and cooked through. Remove the chicken from the skillet and set aside.
Sauté Mushrooms and Garlic: In the same skillet, add the sliced mushrooms. Sauté for about 5 minutes or until they are soft and start to brown. Add the minced garlic and cook for an additional 1-2 minutes, stirring frequently to prevent burning.
Create the Sauce: Pour in the chicken broth and scrape any bits off the bottom of the skillet. Bring to a simmer and let it reduce slightly for about 3-4 minutes.
Add Cream and Seasoning: Lower the heat and stir in the heavy cream and dried thyme. Mix well and let the sauce simmer until it thickens slightly (about 5 minutes). Taste and adjust seasoning with salt and pepper as necessary.
Combine Chicken with Sauce: Return the chicken breasts to the skillet, spoon some sauce over them, and let them simmer in the sauce for another 2-3 minutes to absorb flavors.
Serve: Garnish the dish with chopped fresh parsley before serving.