Place the flour in a shallow bowl. Season the chicken with 1/2 teaspoon salt and 1/4 teaspoon pepper and coat with the flour (tapping off any excess).
Heat the oil in a large skillet over medium-high heat. In batches, cook the chicken until browned, 4 to 5 minutes per side; transfer to a plate.
Add the mushrooms to the skillet and cook, tossing, until browned, 4 to 6 minutes.
Return the chicken to the skillet, add the wine and chicken broth, and simmer until cooked through, about 8 to 10 minutes.
Serve with rice.