Marinade and Sauce:
Combine all marinade and sauce ingredients, except mayonnaise in small bowl; mix well. Divide mixture in half. Place beef steaks and 1/2 of 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 1/2 of 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 (145°F) doneness, turning occasionally. (Do not overcook.) Season steaks with salt, as desired. Serve with sauce.
Nutrition information per serving: 400 Calories; 171 kcal Calories from fat; 19 g Total Fat; 4.5 g Saturated Fat; 0 g Trans Fat; 6.9 g Polyunsaturated Fat; 5.7 g Monounsaturated Fat; 140 mg Cholesterol; 730 mg Sodium; 5 g Total Carbohydrate; 1 g Dietary Fiber; 53 g Protein; 4 mg Iron; 866.2 mg Potassium; 24.4 mg NE Niacin; 1.5 mg Vitamin B6; 4.1 mcg Vitamin B12; 7.6 mg Zinc; 49.9 mcg Selenium; 147.8 mg Choline;