Heat 2 tbsp oil in a large skillet over medium heat. Add cod, season with salt and pepper; cook until just barely opaque and flakey - better to ever-so-slightly undercook the fish than overcook it at this stage, as you'll be cooking it again in the cakes.
Transfer to a bowl and break into large flakes with your hands. Add mashed potato, egg, green onion and enough bread crumbs so that the mixture holds a patty shape (if using leftover mash with milk or other liquid added, you'll definitely need the whole cup of crumbs). Use your hands or a big spoon to work everything together until well-combined.
Place flour on a shallow plate. Form potato mixture into 16 equal patties between the palms of your hands, dredging both sides in the flour and transferring to a baking sheet as you go. Refrigerate fish cakes 20 minutes to firm up.
Wipe out the fish skillet and return to medium heat; add remaining oil. When shimmering hot, add several fish cakes - you'll need to cook them in a few batches to avoid over-crowding the pan. Serve with tartar sauce.