Combine broth, vinegar, and honey.
Melt butter and oil in a large nonstick skillet over low heat.
While butter melts, sprinkle chicken with salt and pepper. Place flour in a shallow dish. Dredge chicken in flour; shake off excess flour.
Increase heat to medium-high; heat 2 minutes or until the butter turns golden brown. Add chicken to pan; cook 4 minutes on each side or until golden brown. Remove chicken from pan; keep warm. Add shallots, and sauté 30 seconds. Add the broth mixture, scraping to loosen browned bits. Bring to a boil, and cook until reduced to 1/2 cup (about 3 minutes). Serve sauce over chicken. Garnish with chopped parsley, if desired.
To assure the chicken gets done, use breasts on the small side, about 4 to 5 ounces each.