For the BBQ Mayonnaise, mix mayonnaise and Bar-B-Q Sauce in small bowl until well blended. Cover. Refrigerate until ready to serve.
For the Onions, mix flour, salt and pepper in large resealable plastic bag. Add onion; toss to coat well. Pour oil into deep fryer, large heavy skillet or saucepan, filling no more than 1/3 full. Heat oil on medium-high heat. Carefully add onion rings to hot oil. Fry 4 to 5 minutes, turning once to brown evenly, until onions rings are golden brown. Drain on paper towels. Set aside.
For the Burgers, mix Seasoning, chili powder and brown sugar in small bowl. Shape ground beef into 4 patties. Coat surface of each patty evenly with the rub mixture. Grill over medium heat 4 to 6 minutes per side or until burgers are cooked through (internal temperature of 160ºF). Add cheese slices to burgers 1 minute before cooking is completed. Grill pickle slices 2 to 3 minutes per side or until grill marks appear. Toast rolls on the grill, open-side down, about 30 seconds, if desired.
Serve burgers on rolls topped with grilled pickles, Crispy Onion Straws and lettuce. Serve with BBQ Mayonnaise.