Season the chicken breasts on both sides with salt and pepper.
Heat the olive oil and butter in a large skillet over medium-high heat.
Sear the chicken breasts for 5 to 7 minutes per side until golden brown and cooked through (internal temperature reaches 165°F or 74°C). Remove the chicken from the skillet and set aside, keeping it warm.
Add the sliced mushrooms to the same skillet. Sauté for 5 to 7 minutes 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 white wine to deglaze the pan, scraping up any browned bits from the bottom. Let the wine reduce by half, about 2 minutes.
Pour in the chicken broth and dried thyme. Bring the mixture to a simmer and cook until the liquid has reduced slightly, about 3 minutes.
Reduce the heat to medium-low. Stir in the heavy cream and Parmesan cheese. Simmer gently until the sauce thickens enough to coat the back of a spoon, about 4 to 6 minutes. Do not boil.
Taste the sauce and adjust seasoning with salt and pepper if needed.
Return the cooked chicken breasts to the skillet, spooning the rich mushroom cream sauce over them.
Garnish with fresh parsley before serving.
Notes
For an extra rich flavor, use chicken stock instead of broth.
If you prefer a thicker sauce, mix 1 teaspoon of cornstarch with 1 tablespoon of cold water, then stir this slurry into the simmering cream sauce.
You can pound the chicken breasts to an even thickness before searing for more consistent cooking.