Add the potatoes, cabbage, green onions, and milk to a large mixing bowl. Stir until smooth. Fold in the cheese and season with salt and pepper.
Chill the mashed potatoes for at least 15 minutes.
Stir in the flour and egg.
Use a medium cookie scoop (2 tablespoons) to divide the potato mixture. Form into patties about ½-inch thick.
Heat a thin layer of oil in a nonstick skillet. Add several potato cakes in a single layer, working in batches if needed, and cook for 2-3 minutes on each side, or until golden and crisp
Blot off excess oil with a towel and transfer to a cooling rack set over a cookie sheet. Serve hot.