Combine flour, basil, salt and pepper in large resealable plastic bag; shake to mix well. Add chicken. Seal and shake to coat; set aside.
Melt 2 tablespoons of the butter in large nonstick skillet over medium-high heat. Add chicken breasts. Place onion around edges of chicken. Cook chicken 4 minutes per side to lightly brown; stir onion occasionally.
Pour broth and vinegar around chicken. Add cherries to pan. Cover; reduce heat and simmer 4 minutes or until meat thermometer registers 170°F. Remove chicken to platter; cover to keep warm.
Increase heat until mixture bubbles; cook 2 to 4 minutes to reduce liquid, stirring occasionally. Reduce heat to low; stir in remaining 1 tablespoon) butter until blended. Serve chicken with sauce, spinach salad and hot crusty bread, if desired.
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