Mash chickpeas and brown rice until thick paste forms. Stir in shallot, garlic, and parsley. Season with salt and pepper. Stir in egg. Form into four 1/2-inch-thick patties.
Heat olive oil in a large skillet over medium-high heat. Add patties and cook until golden brown, about 4 minutes per side. Spread with mustard, top with red onion and roasted red pepper, and wrap in lettuce.