How to Make It
Peel half the potatoes. Cut the peeled potatoes into 1-inch pieces. In a medium saucepan, cover half the potatoes with salted water and bring to a simmer. Cook until the potatoes are tender, about 15 minutes. Drain the cooked potatoes and force through a ricer into a bowl.
Peel the remaining potatoes and grate with a box grater. Wrap the grated potatoes in a kitchen towel and squeeze as much liquid from them as you can. Transfer the grated potatoes to the bowl. Stir in the flour, milk, egg, 3/4 teaspoon salt, 1/2 teaspoon pepper and corned beef.
Form the batter into 1/3-cup-sized patties.
In a large heavy skillet, heat half the butter over medium heat until hot, then cook the patties in batches, turning occasionally, until golden, about 8 minutes per batch. Cook the remaining patties in the remaining butter. Serve.