On a plate, combine flour, 1?2 teaspoon salt and 1/4 teaspoon pepper. Dredge chicken in flour mixture and turn to coat; shake off any excess.
Heat oil in a large nonstick skillet over medium-high heat. Add chicken and cook, flipping once, until golden and cooked through, about 7 minutes; remove to a serving plate and cover to keep warm.
Add onion to skillet; sauté over medium-high heat until lightly browned, about 4 minutes. Add broth, vinegar, thyme and remaining 1?4 teaspoon each salt and pepper. Bring to a boil; cook, stirring often, until onions are tender, about 5 minutes.
Remove skillet from heat and stir in butter until melted; spoon sauce over chicken. Yields 1 chicken breast and about 1/4 cup onion sauce per serving.