Place chicken between sheets of plastic wrap and pound with a meat mallet or heavy skillet until flattened to an even thickness, about 1/2 inch. Season both sides of the chicken with 1/2 teaspoon salt. Place 1/4 cup flour in a shallow glass baking dish and dredge the chicken in it. Discard the excess flour.
Heat 2 teaspoons oil in a large nonstick skillet over medium-high heat. Add the chicken and cook until golden brown, 1 to 2 minutes per side. Transfer to a plate, cover and keep warm.
Heat the remaining 1 teaspoon oil in the pan over medium-high heat. Add shallots and cook, stirring constantly and scraping up any browned bits, until golden brown, 1 to 2 minutes. Sprinkle with the remaining 1 tablespoon flour; stir to coat. Add wine, broth and the remaining 1/2 teaspoon salt; bring to a boil, stirring often.
Return the chicken and any accumulated juices to the pan, reduce heat to a simmer, and cook until heated through and no longer pink in the center, 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.
