In a large bowl, combine the coarsely chopped smoked brisket, diced yellow onion, Worcestershire sauce, smoked paprika, garlic powder, and black pepper. Use your hands to gently mix until everything is evenly distributed, being careful not to overwork the meat.

Divide the brisket mixture into four equal portions. Gently form each portion into patties about 3/4 inch thick, making a slight indentation in the center of each patty to prevent bulging during cooking.

Prepare the tangy sauce by whisking together the mayonnaise, BBQ sauce, and dill pickle relish in a small bowl. Set aside.

Heat a cast-iron skillet or griddle over medium heat. Lightly grease if necessary. Place the patties on the hot surface and cook for 4-5 minutes per side for medium doneness, depending on the thickness of your brisket grind.

During the last minute of cooking, place a slice of sharp cheddar cheese on top of each patty. Cover the skillet briefly to allow the cheese to melt perfectly over the brisket.

While the cheese melts, lightly toast the cut sides of the buns on the same skillet or under the broiler until golden brown.

Assemble the burgers: Spread a generous layer of the tangy sauce on both halves of the toasted buns. Place the cheesy brisket patty on the bottom bun, top with crispy onion rings (if using), and set the top bun in place. Serve immediately.