Pat the chicken breasts dry with paper towels. Season both sides with salt and black pepper.
Heat olive oil in a large skillet over medium-high heat. When hot, add the chicken breasts and cook for 6-7 minutes on each side, until golden brown and cooked through (internal temperature 165°F). Remove chicken from skillet and set aside.
Reduce heat to medium and add butter to the same skillet. Once melted, add the sliced mushrooms and cook, stirring occasionally, for 5 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 chicken broth and scrape the bottom of the skillet to loosen any browned bits. Let it simmer for 2 minutes to reduce slightly.
Stir in the heavy cream and fresh thyme leaves. Cook for another 3-4 minutes until the sauce thickens.
Return the chicken breasts to the skillet and spoon some sauce over them. Cook for 1-2 minutes to warm through.
Sprinkle chopped fresh parsley over the chicken and mushroom sauce before serving.