Sprinkle beef evenly with 1/2 tsp salt and 1/4 tsp pepper. Brown fillets in hot oil in an ovenproof cast-iron skillet over medium-high heat 3 minutes on each side. Remove fillets, reserving dripping in skillet.
Melt butter drippings over medium-high heat. Add yellow and red onion rings, and saute 5 minutes.
Add green onions, shallot, and garlic, and saute 10-15 minutes or until lightly browned. Stir in cognac and broth; cook over medium high heat, stirring constantly, until liquid evaporates (about 5 minutes). Place fillets on top of onion mixture in skillet. Cover with aluminum foil.
Bake at 400 degrees for 15 to 20 minutes or until a meat thermometer inserted into thickest portion of meat registers 140 degrees. (let stand till tempature rises to 145 degrees).
Remove fillets from skillet, reserving onion mixture in skillet; cover fillets loosely, and let stand at room tempature for 10 minutes.
Cook onion mixture over medium heat, stirring constantly, 5 minutes or until liquid evaporates. Add salt and pepper to taste. Serve with fillets.