In a medium bowl, gently mix the first eight ingredients (crab through breadcrumbs) together. Allow mixture to drain in a fine-mesh strainer for a few minutes.
Using your hands, form patties approximately the same size as the sliced bread.
Heat oil in a large non-stick skillet or a large griddle set over medium heat. Once oil is hot and shimmers (just below the smoking point), carefully place the patties one at a time in the pan, being careful not to overcrowd the pan. Cook patties until they are golden brown on the first side, about 4 to 5 minutes. Carefully flip crab cakes and fry on other side until golden brown, about 4 minutes. Remove the patties from the oil once they are golden brown on each side, about 10 minutes total, and place patties directly on the toasted bread that has been layered with lettuce leaves. Fry up any additional patties, adding oil between each cake as necessary.