Melt 4 tablespoons butter and heat 1 tablespoon olive oil in a skillet over medium heat. Cook and stir the mushroom slices in the skillet until tender. Remove pan from heat and set aside.
Melt 1 tablespoon butter and heat 1 tablespoon olive oil in a saucepan over medium-high heat. Stir in the garlic and cook until tender, then gradually whisk in 1 tablespoon flour. Cook, stirring constantly, for 1 minute.
Increase heat to high, and whisk in the beef broth, Marsala, and browning sauce. Season sauce with 1/2 teaspoon salt and 1/4 teaspoon pepper. Bring sauce to a boil, and reduce heat to low. Mix in the cooked mushrooms; you'll use the skillet to cook the chicken. Cover saucepan and remove from heat.
Season chicken with salt and pepper, and dredge chicken breasts in the flour. Over medium heat, melt remaining 3 tablespoons butter in the skillet used to cook the mushrooms. Cook the chicken 2 minutes per side, until browned. Arrange the chicken in the bottom of a 9x13 inch baking dish, and cover with the sauce and mushroom mixture.
Cover baking dish, and bake 25 minutes in the preheated oven, or until chicken juices run clear.