Season both sides of filets with steak seasoning; set aside.
Melt butter in a large skillet over medium-high heat. Add filets and cook for 2 minutes on one side. Turn and cook for 1 minute more. Add garlic and mushrooms and cook for 2 minutes, stirring constantly. Remove filets and set aside on a plate.
Remove pan from stovetop. Tip pan away from you and add cognac. With a (long match or lighter) ignite cognac, keeping pan tipped away from you at all times. When flame has burned out, return pan to stovetop. Stir in steak sauce, cream, and beef broth. Stir thoroughly and bring sauce to a simmer.
Return filets and any accumulated juices to pan. Coat with sauce and warm through. Stir in chives.
Place filets on warm plates. Spoon sauce and mushrooms over filets.