Season steaks to taste with salt and pepper. Melt 2 tablespoons of butter in an ovenproof skillet over medium-high heat. Sear the fillets until browned on both sides, then place into oven and cook to desired doneness, about 13 minutes for medium-rare.
Meanwhile, melt remaining tablespoon of butter in a saucepan over medium-high heat. Stir in mushrooms, and cook until softened, about 2 minutes. Pour in the Marsala, and allow to come to a simmer before adding the balsamic vinegar. Simmer until reduced by half, then remove from heat, season to taste with salt and pepper, and stir in the sliced green onion. Ladle the sauce over steaks to serve.