On a plate, add 3 tablespoons flour, ½ teaspoon salt, and ¼ teaspoon pepper. Combine well. Dredge 4 pieces of chicken in flour mixture to coat; shake off excess.
Heat 2 teaspoons olive oil in a large nonstick skillet over medium-high heat, add 4 pieces chicken and cook, flipping once, until golden brown, about 7 minutes Note: Time will vary depending on thickness of chicken breasts. Remove chicken and cover to keep warm.
Add Vidalia onions to skillet and sauté over medium-high heat for 4 minutes or until lightly browned.
Add to skillet, 1 cup chicken broth, 2 tablespoons balsamic vinegar, 1 tablespoon freshly chopped thyme, ¼ teaspoon salt, and ¼ teaspoon pepper.
Bring to a boil and cook, stirring often, until onions are tender, about 5 minutes.
Remove skillet from heat and stir in 2 teaspoons of butter until melted.
Spoon sauce over chicken and serve.