Heat oven to 350 degrees F.
Place bread crumbs and flour separately in 2 shallow dishes and the eggs in a shallow bowl. Coat chicken in flour and dip in egg, shaking off excess. Coat with bread crumbs and place on a plate.
In a large nonstick skillet, heat 2 tablespoons of the oil over medium-high heat. Sauté chicken 2 to 3 minutes per side, until browned.
In the bottom of a baking dish, mix 1/4 cup of the marinara sauce with 2 tablespoons water. Place chicken in dish and top each with 3 tablespoons sauce and 3 tablespoons cheese. Bake at 350 degrees F, covered, for 15 minutes.
Meanwhile, in a large skillet, simmer broth and garlic, covered, 2 minutes. Add the green beans and mushrooms and simmer, covered, for 5 minutes, until tender. Drain and toss with the remaining teaspoon olive oil and season with salt and pepper. Serve with chicken.