Empty the flour into a shallow dish. Season the chicken with the salt and pepper and then dredge the chicken in the flour until lightly coated. Set aside.
In a large skillet, heat the oil and 1 tablespoon butter over medium-high heat. Working in batches, cook the chicken until browned on both sides and cooked through, about 3 minutes per side. Place cooked chicken on a plate and cover loosely with aluminum foil to keep warm. Set aside.
Reduce heat to medium, add the thyme, mushrooms, and remaining butter and cook until mushrooms have softened and darkened in color, about 6 minutes. Add the wine and broth and cook, stirring, until reduced by half, about 3 minutes. Add salt and pepper to taste. Return the chicken to the pan along with any juices that have accumulated on their resting plate. Top with parsley and serve.
Notes: Martha Stewart Living provided me with the cookbook and enamel cast iron skillet for this post, but all opinions are my own, per usual!