Season the chicken with ½ teaspoon of each salt and pepper. Place the flour in a shallow bowl and dredge the chicken.
Heat the oil and 1 tablespoon of the butter in a large, non-stick frying pan over medium-high heat. Add the chicken and cook for 6 minutes, until it is golden brown. Flip the chicken over and cook for 5-6 minutes more, until it is cooked through. Remove the chicken from the pan.
Add the mushrooms and the remaining 2 tablespoons of butter to the pan and increase the heat to high. They will release a lot of liquid when they begin to cook. After about 10 minutes, once most of the liquid has disappeared from the pan, reduce the heat to medium, season with the remaining ½ teaspoon of salt and pepper, and cook until the mushrooms have browned, about 5-10 minutes more.
Add the wine and thyme leaves to the pan and bring it to a boil. Scrape any brown bits off the bottom of the pan. Add the chicken back to the pan and quickly reheat.
Serve the chicken and mushrooms sprinkled with a little minced parsley.
