For Aioli:

* Whisk all ingredients in small bowl to blend.

* Season dressing to taste with salt and pepper. (Can be prepared 1 day in advance. Cover and refrigerate.)

For Steak:

* Whisk first six ingredients to blend in a 13x9x2-inch baking dish.

* Add steaks to marinade, turn to coat. Cover & refrigerate for at least 8 hours.

For Salad:

* Combine onions, lemon juice and salt in medium bowl. Let stand for one hour, tossing occasionally.

* Prepare grill. Remove steaks from marinade. Sprinkle with salt & pepper.

* Grill 3 minutes per side for medium rare. Transfer steaks to work surface.

* Melt butter if skillet over medium-high heat. Add shallot and mushrooms; saute until well browned, about 7 minutes.

* Add vinegar to skillet and bring to a boil, scraping up browned bits.

* Remove from heat and mix in oil. Season mixture with salt & pepper.

* Combine lettuces, onions, and mushroom mixture in bowl. Toss.

* Divide lettuce mixture among 6 plates. Thinly slice steaks across grain; arrange atop lettuce mixture.

* Drizzle some of Aioli over salads. Pass remaining Aioli separately.