In a large pan boil the potatoes in water to cover until soft. Drain and dry slightly in the hot pan. Mash, adding butter to taste and half-and-half. Continue to mash or whisk, adding more half-and-half as needed, until the potatoes are soft and fluffy but not runny. Add the scallions and season with salt and pepper. Cool the mixture. Add the egg and stir to combine. Chill for several hours or overnight.
Scoop a half-cup measure of the chilled potato mixture and form it into a cake by flattening it on a floured surface. (The cakes will resemble thick hamburger patties.) Dust the other side with flour. Heat about 2 tablespoons of oil in a heavy-duty nonstick skillet and fry several cakes at a time, gently turning once, until nicely browned on both sides.