Combine all marinade/sauce ingredients except mayonnaise in small bowl; mix well. Divide mixture in half. Place beef steaks and half the mixture in food-safe plastic bag; turn steaks to coat. Close bag securely and marinate in refrigerator 6 hours or as long as overnight. Stir mayonnaise into remaining marinade mixture for sauce; cover and refrigerate.
Remove steaks from marinade; discard marinade. Place steaks on grid over medium, ash-covered coals. Grill, covered, 13 to 15 minutes (over medium heat on preheated gas grill, 17 to 19 minutes) for medium rare (145F) doneness, turning occasionally. (Do not overcook.) Season steaks with salt, as desired. Serve with sauce.
Cook's Tip: Recipe can be prepared with 1 beef flank steak (about 1 1/2 to 2 pounds), 1 beef skirt steak (about 1 1/2 pounds) or 1 beef top round steak, cut 1 inch thick (about 1 3/4 pounds). Marinate steak as in recipe above. Prepare recipe as directed above. Grill flank steak on charcoal grill, 11 to 16 minutes (on gas grill, 16 to 21 minutes); grill skirt steak on charcoal grill 7 to 12 minutes (on gas grill, 8 to 12 minutes) for medium rare to medium doneness. Grill top round steak on charcoal grill, 12 to 14 minutes (on gas grill, 16 to 19 minutes) for medium rare doneness; do not overcook. Carve steak across the grain into thin slices.