In a large, heavy-bottomed Dutch oven over medium-low heat, add butter and drizzle olive oil. When butter has melted, add chicken thighs, and cook for about 3-5 minutes. With tongs, flip chicken thighs over to cook on other side. Cook an additional 3-5 minutes. Using tongs, remove chicken to a plate and set aside.
Add whole garlic cloves and mushrooms to Dutch oven. Stirring constantly, cook garlic and mushrooms until lightly browned, taking care not to burst garlic. Remove from Dutch oven to a small bowl and set aside.
Add chicken stock to the bottom of the Dutch oven, stirring to remove any browned bits that have stuck to the bottom of the pan. As stirring, add half and half to the chicken stock.
Add back chicken, then garlic and mushrooms. Add thyme.
Cover and allow to simmer over medium-low heat for 45 minutes, or until the juices run clear when the meat of the chicken thigh is pierced with a knife.
Serve warm.
Notes
Serve chicken with a spoon of garlic and mushroom sauce drizzled over the top. Also delicious served with rice, mashed potatoes, or pasta.