Scrub and rinse potatoes. Do not peel. Place in medium-sized saucepan. Cover with water. Simmer over medium-high heat until fork tender. Instant potatoes will not work in this recipe.
Drain. Allow to cool. Remove skins. Mash until semi-smooth.
In a medium-sized bowl, thoroughly combine mashed potatoes (do not use instant), melted butter, beaten egg, self-rising flour, chopped green onions, garlic powder (if using), salt, black pepper, and shredded cheddar cheese (if using).
Scoop a generous tablespoon of potato mixture to form a ball. Lightly flatten to form a patty. Repeat until all potato mixture is used. Place patties on a plate and set aside.
Pour 3 tablespoons of vegetable oil (or olive oil) in a 10" cast iron skillet or preferred cookware. Warm over low heat until sizzling hot (important step).
Use spatula, to scoop patties from plate and place in hot/greased skillet.
Place 4 patties in hot/greased skillet. Fry over medium-high heat for 4-5 minutes. Flip over and repeat. Use slotted spoon to remove fried patties and place on paper towels to drain.
Pour 3 tablespoons of vegetable oil (or olive oil) in hot skillet. Wait 3 minutes before adding remaining patties (so oil will get hot). Repeat as previously instructed.
Serve warm or hot. Top with sour cream and/or sprinkle with grated Parmesan cheese (optional).