Peel onions and slice crosswise into medium-thick rounds.
Separate all the rounds into individual rings.
Put 2 cups all-purpose flour into a large bowl and season with one tablespoon salt and one tablespoon cayenne.
Dredge the onion rings in the seasoned flour and set aside.
Working in batches to avoid crowding the skillet, fry the onion rings in a large deep skillet of hot vegetable oil over medium-high heat, turning them as they brown, until golden and crisp, 3- 5 minutes per batch.
Drain the onion rings on paper towels and season to your taste with salt while still warm.
Grill burgers according to directions on package.
Place burgers on buns and top each with a tablespoon of barbeque sauce.
Mound each burger with fried sweet onion rings and serve.
Thaw burgers and divide each one into 3 or 4 equal parts. Shape into mini patties and grill. Proceed with above directions using slider buns.