Heat 2 tbsp oil in a large skillet over medium heat. Add cod, season with salt and pepper. Cook until just barely opaque and flaky. It's 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 fish 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. Cook each side for 3-5 minutes, or until golden brown and crisp. Adjust the heat as needed to prevent burning.
To make tartar sauce:
Combine all ingredients in a medium bowl, whisking together with a fork. Season with salt to taste.Serve fish cakes with the tartar sauce.