Run salmon through a meat grinder on medium setting. If you don't have one of these, ask your butcher to, but ask for it to be clean before he does.
Combine ground salmon, 1/4 cup breadcrumbs, salt and pepper, and lemon. Form into 4 patties, about 1 1/4 inch thick.
Insert a probe thermometer into one of the cakes. Place on a greased baking pan and bake at 350 degrees F until internal temperature reaches 130 degrees F.
While the salmon cooks, heat olive oil in a cast iron skillet to 375 degrees F. Once salmon is parcooked, dredge into more breadcrumbs then immediately transfer to oil. Cook until golden brown on both sides.