Make the dressing: Combine the mayo, ketchup, pickle relish, mustard, vinegar, and salt in a small bowl. Cover and refrigerate until it is needed, at least for 30 minutes.

Make the black bean patties: Pulse the black beans in a food processor until the beans have a mashed texture. Transfer to a bowl and add the panko, egg, miso, salt, paprika, and garlic powder. Stir until well combined, then set aside for 20 minutes.

Cook the onions: Heat the oil in a well-seasoned griddle or cast-iron pan over medium heat. Add the onion and cook, stirring occasionally, until the onion pieces are tender and starting to brown, 14 to 16 minutes. Transfer to a small bowl and set aside.

Cook the patties and serve: Return the griddle to medium heat and drizzle liberally with avocado oil. Lightly wet your hands and form the black bean mixture into eight balls that are slightly larger than golf balls.

Place the black bean ball on the hot griddle. Lay a piece of parchment paper on top, and using the back of a spatula, and your hand, press until the patty is flat and only ¼ inch (6 mm) in thickness. Repeat with as many patties as you can fit on your griddle, working in batches as needed. Cook for 2 to 3 minutes on each side, adding oil to the pan as needed to maintain a thin layer for frying. Once the bottom of each patty is quite brown and crisp, flip and add a slice of cheddar cheese. Cook until the second side is brown and crisp. Transfer the patties to a plate and repeat until all patties are cooked, adding more oil if needed.

Lightly toast the buns on the griddle. Spread a small bit of sauce on the bottom bun, then layer with two patties, lettuce, tomato, onions, and pickles if desired. Spread a bit more sauce on the top bun and add to the burger.