Season chicken breasts with salt and black pepper.
Mix flour, salt, and black pepper together in a shallow bowl. Dredge chicken in flour mixture, shaking off excess.
Heat oil in a large skillet over medium-high heat. Add chicken; cook until browned, about 3 minutes per side. Remove chicken from the skillet. Add onions and garlic; saute until golden, about 3 minutes.
Return chicken to the skillet. Add porcini mushrooms; cook and stir until starting to soften, about 3 minutes. Stir in chicken broth, balsamic vinegar, and white wine. Cover and continue cooking over medium-low heat, turning chicken occasionally, until no longer pink in the center and coated with sauce, about 10 minutes.