Sprinkle beef evenly with 1/2 teaspoon salt and 1/4 teaspoon pepper. Brown fillets in hot oil in an ovenproof or cast-iron skillet over medium-high heat 3 minutes on each side. Remove fillets, reserving drippings in skillet.
Melt butter in drippings over medium-high heat. Add yellow and red onion rings, and sauté 5 minutes.
Add green onions, shallots, and garlic, and sauté 10 to 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° for 15 to 20 minutes or until a meat thermometer inserted into thickest portion of meat registers 135° (medium rare).
Remove fillets from skillet, reserving onion mixture in skillet; cover fillets loosely, and let stand at room temperature 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.
*1/4 cup red wine or beef broth may be substituted for cognac.