Mince the garlic cloves and onion finely, keeping them separate. Crush the toasted walnuts into a coarse meal using a food processor or by hand—you want texture, not powder. Drain and rinse the garbanzo beans, then place them in a large mixing bowl and mash about 90% of them with a fork, leaving some beans whole for texture. In a separate small bowl, whisk together the flour, cumin, garlic powder, onion powder, smoked paprika, salt, and pepper to combine all dry ingredients evenly.
Heat 1 1/2 tablespoons of olive oil in a skillet over medium heat. Add the minced onion and sauté for about 3 minutes until softened and starting to become translucent. Add the minced garlic and cook for another 2 minutes until fragrant. I like to cook the aromatics this way rather than using them raw because it develops deeper, more rounded flavors in the burger patties. Remove from heat and let cool slightly before adding to the bean mixture.
Add the cooked rice, sautéed onion and garlic mixture, walnuts, and dry ingredient mixture from Steps 1 and 2 to the bowl with the mashed beans. Add the soy sauce and egg white, then mix thoroughly with your hands or a wooden spoon, breaking up any rice clumps as you go. The mixture should hold together when pressed but still have some texture. Once combined, divide the mixture into 6 equal portions and shape each into a patty about 1 inch thick, being gentle to avoid overworking them. Brush both sides lightly with the remaining 1 1/2 tablespoons of olive oil.
Preheat your grill to medium heat (around 375–400°F if using a thermometer). Once hot, place the burger patties on the grill and cook for 10–12 minutes without moving them. This initial phase is crucial for developing a nice crust and allowing the interior to hold together. The slight char will add depth to the flavor, so resist the urge to flip early.
Carefully flip each burger patty and immediately spread about 1 tablespoon of BBQ sauce on the cooked side. Grill for another 10–12 minutes until the second side is nicely browned and the burgers feel firm when gently pressed. I find that applying the sauce right after flipping gives it time to caramelize slightly without burning. The burgers are done when they hold together well and have a pleasant grilled exterior.
Lightly toast the brioche or sesame buns if desired for better texture. Place a cooked burger patty on the bottom half of each bun, then top with a dollop of fresh guacamole, a slice of tomato, and a leaf of crisp green leaf lettuce. Cap with the top bun and serve immediately while the burgers are still warm.