Pat the chicken breasts dry with paper towels. Pound between sheets of plastic wrap to an even thickness.
Combine 4 tablespoons flour with 1/2 teaspoon salt and the pepper. Dredge chicken breasts to coat.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken to the hot oil and brown for about 3 minutes on each side. Add sliced mushrooms and sauté for 1 minute longer.
Meanwhile, in a small bowl, combine the garlic, tomatoes, wine, and chicken broth; stir in the remaining 2 tablespoons of flour until well blended. Pour the mixture over the chicken and mushrooms; stir to combine and bring to a simmer. Reduce heat to low and simmer for about 12 to 15 minutes, or until the chicken is thoroughly cooked.