In the bowl of a food processor, pulse 4 slices stale or toasted whole wheat sandwich bread to form breadcrumbs. Pour into a bowl and set aside.
Heat 1 tablespoon olive or canola oil in a large skillet set over medium heat. Add 1 medium onion, chopped, and sauté until soft. Add 2 cloves garlic, minced, 2 tablespoons minced jalapeno pepper (membranes and seeds removed – unless you like more heat), and 1 teaspoon ground cumin. Saute for additional 2 minutes.
Scrape the mixture into the food processor. Add 1 (15 oz.) can black beans, drained and rinsed, 1 ¼ cups cooked brown rice, 2 teaspoon fresh lime juice, ½ teaspoon kosher salt, and ½ teaspoon freshly ground black pepper. Pulse until the beans are chopped, but not pureed.
Scrape the black bean mixture into a large bowl. Add ½ of the breadcrumbs and ¼ cup chopped cilantro. Using a wooden spoon, stir the mixture until combined.
Divide the mixture into 4 equal portions and form into patties. If the mixture is too moist to form patties, add more breadcrumbs. Place the patties on a baking sheet lined with plastic wrap and cover with additional plastic wrap. Refrigerate for at least 1/2 hour.
Heat 1 tablespoon canola oil in a large skillet (preferably not nonstick, so a good crust can form) set over medium heat. Add the burgers and cook for 4 minutes per side.
In a small bowl, combine ½ cup mayonnaise, 2 teaspoons fresh lime juice, ½ teaspoon ground cumin, 2 tablespoons finely chopped cilantro, and ¼ teaspoon freshly ground black pepper. Stir well.
Lightly toast 4 whole wheat buns. Spread 2 tablespoons cilantro-lime mayo on each bun, and top each with a black bean patty, 2 tablespoons crumbled cotija cheese (found in most supermarkets), 1 slice large tomato, and 3 slices avocado. Serve.