Stir potatoes, beaten egg, cheddar, and green onions in a bowl until mostly smooth.
Season with garlic powder, onion powder, salt, and pepper; mix to combine.
Fold in the flour until a soft, scoopable mixture forms; add a spoonful more if very sticky.
Scoop about 1/4 cup per cake, then pat into 1/2-inch-thick patties. Lightly dust with flour.
Heat oil and butter in a large skillet over medium heat until shimmering.
Pan-fry cakes 3–4 minutes per side until deep golden and hot inside. Cook in batches as needed.