Serve these vegan burgers with pickles and fries on the side and a sundae for dessert.
Fill a small- to medium-sized saucepan two-thirds full with heavily salted water, cover, and bring to a boil over high heat. Once the water is boiling, stir in the rice and cook, stirring occasionally, until tender, about 40 minutes. Pour into a colander set in the sink and let cool to room temperature, about 15 minutes (the rice should have doubled in volume, to a bit more than 1/2 cup).
Meanwhile, heat 1 tablespoon oil in a small, heavy, nonstick sauté pan over medium-high heat. When hot, add the garlic and sauté until slightly softened, 1 to 2 minutes (watch carefully to prevent burning). Immediately add the mushrooms and sauté, stirring, until tender and browned, about 5 minutes. Set aside off the heat to cool (you should yield 1/2 cup).
Add the cooked rice, the garlic-mushroom mixture, 1/4 teaspoon salt, the chickpeas, carrots, parsley, ketchup, onions, mustard, soy sauce, chipotle liquid, and black pepper to the bowl of a food processor. Purée until the burger mixture is combined well, about 40 seconds.
Mix the panko and the remaining 1/4 teaspoon salt in a large bowl. Add the burger mixture and mix well with your hands until combined. Shape into 6 patties and flatten each one to about 1/2 inch thick, using your fingers to make an indentation in the center of each.
Heat 2 tablespoons oil in a 10-inch, heavy, nonstick sauté pan over medium-high heat. When the oil is hot, add 4 patties and cook until the first side is browned, about 3 minutes. Flip over and cover each patty with a slice of cheese. Cover the pan and reduce the heat to medium. Continue cooking until the cheese has melted and the other side is browned, about another 3 minutes. Transfer to a plate. Raise the heat to medium high, and add the remaining 2 tablespoons of oil and remaining 2 patties to the pan. Repeat the cooking process, turning the burgers over after about 3 minutes, then adding the cheese, covering the pan, reducing the heat to medium, and cooking until the cheese has melted.
Place a patty on half of each bun and top with a tomato slice and some avocado slices. Serve immediately, passing ketchup and mustard at the table.