Heat oil in an oven-safe skillet on high heat. Place chicken, skin side down, into the pan. Cook until fat renders, about 7 minutes.
Transfer the pan to the oven. Cook 5 minutes. Flip chicken over and cook another 5-7 minutes. Remove chicken to a separate plate.
Place the skillet back onto the stove over medium heat. Melt the butter in the pan. Add garlic, mushrooms, salt and pepper. Cook until mushrooms release their liquid, about 5-7 minutes.
Pour in the chicken broth. Increase the heat to medium-high, scraping any browned bits off the bottom of the pan.
Lower the heat again and pour in the cream. Cook 1-2 minutes, until mixture thickens slightly. Spoon cream sauce over the chicken thighs. Serve warm.