Rinse the chicken and pat dry. Sprinkle with salt and pepper. Dredge the chicken in a mixture of flour, salt, and pepper; shake off any excess flour. Cook the fillets in the olive oil in a saute pan over medium-high heat for 3 minutes on 1 side or until brown. Add the garlic. Turn the chicken over; sprinkle with the mushrooms. Cook for 3 minutes, shaking the saute pan to redistribute the mushrooms so they will cook evenly. Stir in the balsamic vinegar, chicken broth, and wine. Cook, covered, over medium-low heat for 10 minutes, turning the chicken occasionally.