Pat the chicken breasts dry with paper towels and season both sides generously with salt, pepper, and dried thyme.
Heat the olive oil in your skillet over medium-high heat. When hot, add the chicken breasts and cook for about 5-6 minutes on each side until golden brown and cooked through. Remove the chicken from the skillet and set aside, loosely covered with foil to keep warm.
In the same skillet, add the sliced mushrooms. Sauté until they release their moisture and start to brown, about 5 minutes. Add the minced garlic and cook for another 1-2 minutes until fragrant, stirring frequently to prevent burning.
Pour the heavy cream into the skillet with the mushrooms and garlic. Stir well to combine, scraping up any browned bits from the bottom of the pan. Let the sauce simmer gently until it thickens slightly, around 3-5 minutes. Taste and adjust seasoning with salt and pepper.
Return the chicken breasts to the skillet, nestling them into the creamy mushroom sauce. Spoon some sauce over the top and cook together for another 2 minutes to meld the flavors. Sprinkle with chopped parsley before serving.