Season chicken with salt and pepper. Coat with 1/4 cup flour.
Heat oil and 1 tablespoon butter in an ovenproof skillet over medium-high heat. Add chicken and saute until lightly browned on both sides, about 5 minutes. Add garlic; saute until fragrant, about 30 seconds.
Pour wine into skillet and bring to a boil, scraping the browned bits of food off of the bottom of the pan with a wooden spoon. Simmer until wine has reduced by half, about 5 minutes. Add broth, mushrooms, and parsley. Remove from heat.
Divide remaining butter into 3 pieces; coat with the remaining flour and add to skillet. Season with salt and pepper and place skillet in oven.
Bake in the preheated oven until chicken is no longer pink in the center and the juices run clear, 25 to 30 minutes. An instant-read thermometer inserted into the center should read at least 165 degrees F (74 degrees C).