In a low wide bowl, combine flour, ¼ teaspoon salt, and ¼ teaspoon of pepper. Dredge the chicken in the flour mixture, shaking off the excess. In a large nonstick skillet, heat the oil until hot but not smoking over medium-heat. Add the chicken and cook until golden brown, about 2 minutes per side. Transfer the chicken to a plate. Set aside.
Add green onions and garlic to the pan and cook until the green onions are softened, about 1 minute. Add the vinegar, increase the heat to high and cook until slightly evaporated, about 1 minute. Add the remaining ingredients, plus the remaining ¼ teaspoon salt and ¼ teaspoon pepper. Bring to a boil, add the chicken back to the pan, reduce to a simmer; cover and cook until the chicken is cooked through, about 8 minutes. Divide the chicken and sauce among 4 bowls and serve.
Notes
If you are preparing this recipe as gluten free, use GF all-purpose flour in place of regular, and be sure to use a brand of chicken broth that is designated as gluten free.