Season chicken well on both sides with salt and pepper. Pre-heat a cast iron skillet or large Dutch oven over medium high heat. Add 2 tbsp. butter and olive oil. When the butter melts, add half of the chicken, skin side down. Brown well on all sides, about 10 minutes. Remove chicken, set aside, and repeat with remaining chicken.
Reduce heat to medium, and add mirepoix (onion, carrot, and celery) to pot, scraping up any browned bits with a wooden spoon. Sauté, stirring occasionally, until soft , 8 to 10 minutes.
Add cremini mushrooms and cook about 4-5 minutes, stirring occasionally, until the mushrooms darken and begin to release liquid. Stir in flour, and cook for about 1 minute, or until the flour is absorbed by the vegetables. Make sure to scrap up any of the browned bits with a wooden spoon.
Add wine, bring to a boil, stirring until liquid thickens, about 1 minute. Add broth and stir well to incorporate.
Return chicken to the pot, skin side up, pour any juices that hae accumulated on the plate into a pot. Add thyme and bay leaf. Bring to a boil, reduce to a simmer, cook partially covered until the internal temperature of the chicken reaches 165 degrees, about 25-30 minutes. Remove chicken to a plate and simmer the sauce uncovered, until reduced slightly, about 5 minutes. Remove and discard herbs.
Stir in cream, lemon juice, and remaining butter. Return chicken to sauce and serve.