Place the flour in a shallow bowl. Season the chicken with ½ teaspoon salt and ¼ 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.
Meanwhile, bring a large saucepan of water to a boil; add 1 tablespoon salt. Add the green beans and cook until just tender, 4 to 5 minutes. Drain and run under cold water to cool. Serve with the chicken and top with chives.