I found this recipe in a grocery store checkout cookbook (Pillsbury Casseroles and Slow Cooker, Jan 2011). "Does it have chili powder", my husband asked suspiciously. "Yes, but just a little bit" (maybe I overuse chili powder, lol). But, when all was said and done, he went back for more and asked if we could make it again! We're going to try to make it regular hamburger bun size, though. We'll see!
In a 2 quart saucepan, heat 4 cups water to boiling. Stir in cabbage. Remove from heat, let stand 5 minutes. Meanwhile, in medium bowl, mix beef, egg, cumin, seasoned salt, chili powder, onion, cilantro, and lime juice. Divide mixture into 12 portions; shape into 12 patties, about 3" by 3/8" thick.
Drain cabbage and squeeze dry; set aside. In small bowl, stir mayonnaise and ketchup; set aside.
Spray 10" skillet with cooking spray; heat over medium heat. Cook 6 patties in skillet 5 minutes; turn. Top each patty with 1 tablespoon of the cheese. Cook 2 minutes longer or until meat thermometer inserted in center of patties reads 160*F. Remove burgers; cover to keep warm. Repeat with remaining patties and cheese.
To serve, spread 2 teaspoons mayonnaise mixture on each roll bottom; top with 1 burger, 1 tomato slice, 1 avocado slice, and a scant 1/4 cup cabbage. Cover with top half of rolls; secure with toothpicks.