In a food processor, add the black beans, grated carrot, chopped bell pepper, chopped red onion, minced garlic, and parsley. Pulse until the mixture is well combined but still has some texture.
Transfer the mixture to a large bowl. Add the cooked quinoa, breadcrumbs, soy sauce, smoked paprika, cumin, salt, black pepper, and beaten egg. Mix until well combined. If the mixture feels too wet, add more breadcrumbs; if too dry, add a little water.
Divide the mixture into 4 equal portions and shape each portion into a patty about ½ inch thick.
Preheat a grill pan over medium-high heat and brush it lightly with olive oil. Place the patties on the grill pan and cook for about 4-5 minutes per side, or until nicely browned and heated through. If using cheddar cheese, place a slice on each patty during the last minute of cooking to melt.
Toast the burger buns on the grill pan for about 1 minute, or until lightly browned. Place each veggie patty on a bun and add your desired toppings: lettuce, tomato slices, red onion slices, and avocado slices, mayo.