Heat oil in a large skillet over medium-high heat. Add onion and cook, stirring, until starting to brown, 3 to 5 minutes. Add thyme, flour, salt and pepper and stir to coat. Cook, stirring, for 1 minute more. Add mushrooms, broth and wine and bring to a boil. Reduce heat to maintain a lively simmer and cook, stirring, until the sauce has thickened and the mushrooms are cooked through, 12 to 15 minutes more. Remove from heat and stir in butter.