Bring the brown rice and water to a boil in a saucepan over high heat. Reduce the heat to medium-low, cover, and simmer until the rice is tender, and the liquid has been absorbed, 45 to 50 minutes.
Preheat an outdoor grill for high heat. Lightly oil a sheet of aluminum foil. Mash black beans in a large bowl with a fork until thick and pasty; set aside.
Place the bell pepper, onion, mushrooms, and garlic in the bowl of a food processor, and chop finely. Stir the bell pepper mixture into the mashed black beans. Place the brown rice and mozzarella cheese in the food processor, and process until combined. Stir the mixture into the black beans.
Whisk together the eggs, chili powder, cumin, garlic salt, and hot sauce. Stir the egg mixture into the black bean mixture. Stir in the bread crumbs, adding additional bread crumbs as needed until the mixture is sticky and holds together. Divide into 6 large patties.
Place patties onto the prepared foil, and grill until browned and heated through, about 8 minutes per side.