In a large bowl, stir together the mashed potatoes, cheese,green onions, egg and 1 tablespoon flour until combined. Divide the mixture into 6 portions. Roll each portion into a compact ball then flatten it into a pancake about a 1/2-inch-thick.
Place the remaining flour in a shallow dish and dredge each pancake in the flour.
Heat 3 to 4 tablespoons of vegetable oil in a large sauté pan over medium heat. (Add enough oil to thoroughly coat the bottom of the pan.)
Fry the pancakes, in batches, until they're golden brown and crispy on both sides, 3 to 4 minutes. Add more oil to the pan as needed between batches.Place the pancakes on a paper towel-lined plate and immediately sprinkle them with salt.