Melt butter in a skillet. Add green onions, mushrooms and thyme. Cook, uncovered, 3-5 minutes, or until onions are tender, stirring occasionally. Remove from heat. Remove about 1/3 of the mushroom mixture to serve over the finished steaks. Set aside. Stir cheese into remaining mixture.

Cut a3-inch wide pocket in the side of each tenderloin filet, cutting to but not through the other side. Spoon half the cheese-mushroom mixture into each pocket. Sprinkle meat lightly with salt and pepper. Wrap 2 slices of bacon around each steak; secure with toothpicks.

Place on unheated rack of broiler pan. Broil 4 to 5 inches from heat 20 to 24 minutes, or until an instant-read thermometer inserted in center of meat and stuffing registers at 145°F for medium rare and 160°F for medium, turning steaks halfway through cooking time. Transfer to plates; top with reserved mushroom mixture.