In a large skillet, heat 1 tablespoon of the oil over medium heat. Season the chicken with 1/4 teaspoon salt, and 1/4 teaspoon pepper. Add the chicken to the skillet. And cook until golden brown on one side, about 5 minutes. Flip the chicken and cook until golden on the second side, about 3 minutes. Transfer to a plate.
Add the remaining 1 tablespoon oil and garlic to the skillet and cook, stirring until the cloves are fragrant and lightly browned, about 2 minutes. Add the mushrooms and thyme and continue to cook, stirring until the mushrooms are tender, about 5 minutes. Pour in the wine and use a wooden spoon to scrape up any browned bits from the bottom of the pan. Stir in the broth and bring to a simmer. Cook until the sauce is reduced by half, about 5-7 minutes.
Season the sauce with the remaining 1/4 teaspoon salt and 1/4 teaspoon pepper. Return the chicken and any accumulated juices to the pan, add 2 tablespoons of the parsley, and stir to coat the chicken all over.
Remove the garlic cloves and thyme sprigs. Sprinkle the remaining tablespoon of parsley over the chicken just before serving.