Preheat the oven to 350. Spray a 9 x 13 baking dish with cooking spray.
Heat the olive oil in a large skillet over medium heat. Add the mushrooms and garlic to the skillet, and sprinkle with salt and pepper. Cook, stirring frequently, until the mushrooms are soft. Add the mushroom soup and chicken broth to the skillet, and cook until well combined and heated through.
To prepare the chicken, cut the chicken breasts in half horizontally to form thin fillets. In a shallow dish, whisk together the eggs and milk. In another shallow dish, whisk together the bread crumbs, garlic powder, salt, and pepper.
Dip the chicken pieces in the egg mixture, then coat with the bread crumbs on both sides. Heat the butter in a large skillet over medium heat. Add the breaded chicken to the skillet and cook 3-4 minutes per side, until golden brown on both sides.
Pour half of the mushroom sauce into the prepared baking dish, then layer the chicken pieces on top. Pour the remaining mushroom sauce over the top of the chicken, and cover with the slices of Swiss cheese.
Bake 35-40 minutes, until the chicken is cooked through. Serve hot with noodles, rice, or mashed potatoes.