To prepare chicken, place flour in a shallow dish. Sprinkle chicken evenly with 1/4 teaspoon kosher salt; dredge chicken in flour. Heat 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Add chicken to pan; cook 6 minutes on each side or until chicken is done and lightly browned. Remove from pan; keep warm. Wipe pan with a paper towel.
To prepare chutney, heat 1 teaspoon olive oil in pan over medium-high heat; add onion, ginger, and garlic to pan. Cook 4 minutes or just until tender. Add tomato and remaining ingredients; bring to a boil. Cover, reduce heat, and simmer 20 minutes. Uncover and cook 6 minutes or until slightly thickened. Serve chutney with chicken.