PREP: Grate an onion on the large holes of a cheese grater. Dice a bell pepper and mushrooms (the food processor will break down further, but helps if they're diced first).
FOOD PROCESSOR: Add 1/2 cup cashews to a large (12-cup) food processor and pulse 10 times or until coarsely chopped (no smaller). Add in red onion, red pepper, mushrooms, black beans, 1 tbsp. garlic, 2-1/2 tsp each of chili powder, paprika, and cumin, 1/2 teaspoon each of garlic powder, salt, and pepper. Pulse 5 times. Scrape down the sides, stirring ingredients, and pulse another 5 times. Scrape sides and pulse 5 more times. Ingredients should be well incorporated. Add in 1/4 cup ketchup, 1 tbsp. Worcestershire sauce, and 1 egg. Pulse to combine, stirring in between, about 10 pulses. Add in 1-1/4 cup Panko, and 1 cup cheese. Pulse to combine, stirring in between about 10 more pulses. Finally, add cooked 3/4 cup rice. Pulse 3 times, scrape edges and stir and pulse 3 more times (we're only pulsing to incorporate the rice--we don't want to break it down or you'll get gummy burgers!). (Trouble-shooting: crumbly patties? Pulse longer. Overly wet? Add some more Panko!)
FORM PATTIES: Pack a 1/2-cup measuring cup with the mixture and plop onto a parchment-paper or foil-lined sheet pan. Repeat to get 9-10 mounds of the mixture. (Divide any remaining mixture from food processor among the mounds to evenly disperse.) Lightly spritz your hands with cooking oil spray and then shape each mound into a patty by pressing it down on the sheet pan and shaping it with your hands. Each patty should be about 3/4-inch thick and about 4 inches wide (or slightly wider than the width of the buns). Place shaped patties in the freezer for 25-30 minutes or until firm enough to gently pick up off the tray with a metal spatula.
COOK ON STOVETOP: Heat a drizzle of oil (about 2 tbsp) over medium-high heat. Once oil is hot, add 3 patties straight from the freezer to the pan and cook until browned, golden, and crispy about 3-4 minutes. Carefully flip then cook the other side for another 3-4 minutes or until deeply golden. Transfer to a paper-towel lined plate. Cook the next batch following the same method. Don't crowd the pan with too many burgers at a time.
SAUCE: Combine all of the sauce ingredients in a small bowl. Stir until combined and smooth. Taste and adjust seasonings to personal preference.
ASSEMBLE: Toast buns (Note 6). Place the bottom of the bun on a plate and add mayo or burger sauce. Layer with lettuce, veggie patty, avocado, tomato, and onion. Add mayo or burger sauce to the top bun. Sandwich together and enjoy promptly.