For the special sauce: Stir together the mayonnaise, ketchup, pickle relish, pickle juice, garlic powder, paprika and a pinch each of salt and pepper and set aside.
For the burgers: Add the black beans, beets, garlic powder, onion powder and 1/2 teaspoon salt to a food processor and pulse until coarsely chopped. Add the panko, rice, Worcestershire sauce and liquid smoke and pulse just until the mixture starts to come together. Transfer the mixture to a medium bowl and stir until thoroughly combined. Using wet hands, form 8 patties and place on a plate.
Heat a cast-iron pan over medium-high heat and add 2 tablespoons of the oil. Add the patties and cook for about 4 minutes. Flip the patties, top with the cheese, if you like, and add the remaining tablespoon of oil to the pan. Cook until patties are hot and the cheese has melted, about 4 minutes more.
To assemble, spread each bottom bun with a thin layer of the special sauce and top with some of the pickles and onions. Cover with a patty, followed with some lettuce and repeat the process with the remaining sauce, pickles, onions, patties and lettuce. Top with the top bun.