Stir together the mashed potatoes, cooked cabbage, green onions, cheddar, egg, flour, salt, and pepper until evenly combined.

With floured hands, shape the mixture into 12 small patties about 2 inches across and 1/2 inch thick.

Melt half the butter in a large skillet over medium heat.

Working in batches, fry the patties for 3 to 4 minutes per side, until deep golden brown and crisp at the edges, adding the remaining butter for the second batch.

Transfer to a paper towel lined plate to drain briefly.

Serve warm with a small dollop of sour cream on each.