In a large bowl, combine eggs, onion, bread crumbs, beer, Worcestershire sauce, thyme, dry mustard, and cayenne pepper. Add ground beef; mix well. Shape mixture into twelve 3/4-inch-thick patties.
For a charcoal grill, grill patties on the rack of an uncovered grill directly over medium coals for 14 to 18 minutes or until done (160 degrees F), turning once halfway through grilling. (For a gas grill, preheat grill. Reduce heat to medium. Place patties on grill rack over heat. Cover and grill.)
Serve burgers on toasted buns. Spoon Cajun Beer Sauce over burgers. If desired, serve with lettuce and tomatoes.
Cajun Beer Sauce
In a medium saucepan, cook 1/2 cup chopped onion (1 medium) and 1/2 cup chopped green and/or red sweet pepper (1 small) in 1 tablespoon hot vegetable oil over medium heat about 5 minutes or until tender. In a small bowl, combine 1 cup Abita beer or desired beer, 1 cup cold water, 2 tablespoons cornstarch, and 2 teaspoons Cajun seasoning. Add to onion mixture in saucepan. Cook and stir until thickened and bubbly. Cook and stir for 2 minutes more.