Place chicken between sheets of plastic wrap and pound with a meat mallet or heavy skillet until flattened. Season both sides with ½ teaspoon salt. Place ¼ cup flour in a small glass baking dish and drag the chicken in it. Discard excess flour.
Heat 2 teaspoons oil in large skillet over medium-high heat. Add the chicken and cook until golden brown. Transfer to a plate, cover and keep warm.
Heat remaining 1 teaspoon oil in the pan over medium-high heat. Sprinkle with the remaining 1 tablespoon flour; stir to coat. Add wine, broth, and the remaining ½ teaspoon salt; bring to a boil, stirring often.
Return chicken and any accumulated juices to the pan, reduce heat to a simmer, and cook until heated through, about 6 minutes. Stir in sour cream and mustard until smooth; turn the chicken to coat with the sauce. Stir in chives and serve immediately.