1. Heat a grill. When hot, grill the onions for about 2 minutes on each side, or until they are tender but crunchy.
2. If you don't have a grill, blanch the onions in boiling salted water.
3. Drain and keep warm.
4. Season the steaks on both sides with salt and pepper.
5. Reserve.
6. In a heavy skillet, heat 2 tablespoons of butter until bubbling slightly.
7. Cook the shiitakes over high heat until they begin to brown slightly.
8. Season lightly with salt and pepper.
9. Remove from the pan and keep warm.
10. Carefully pour the Cognac or brandy into the hot skillet and ignite.
11. Let the flame die out naturally.
12. Reduce the Cognac to 2 tablespoons, scraping any loose bits that cling to the bottom of the pan.
13. Stir in the reduced beef stock.
14. Reduce the sauce until slightly thickened, but still very light.
15. Return the mushrooms and their juice to the sauce and reduce more if necessary.
16. Correct the seasonings.
17. Stir in the remaining 2 tablespoons butter, a little at a time.
18. Stir in the enoki mushrooms.
19. Set the sauce aside and keep warm.
20. Grill the steaks on the preheated grill for 3 minutes on each side, or saute them in a very hot skillet with a little oil for 3 minutes on each side, or until the meat is medium rare.
21. Remove from the heat.
22. If necessary, reheat the sauce very gently.

Yield: 4 servings