Generously salt and pepper the chicken thighs. Dip them into the flour and shake off any excess.
In a large skillet with a fitted lid, add olive oil and 2 tablespoons butter over medium-high heat. When the butter has melted, and the oil is hot but not smoking add the chicken skin side down. Sauté until golden brown. Turn over the chicken.
Meanwhile, in another skillet add the remainder 2 tablespoons butter over medium-high. When the butter has melted, and bubbling, add the mushrooms and sauté until golden.
Season with salt and pepper and add the mushrooms to the chicken thighs.
Add white wine to the skillet. Lower the heat to low, cover, and simmer for 30 minutes.