You'll need to adjust the number of eggs used to accommodate the moisture in the mashed potatoes. The egg serves as a binder. You want the patty to be somewhat sticky so it will stay together. It should be thick and hold together well before you put it in the pan to cook. If you're mashed potatoes are very creamy you might need to add a little all-purpose flour to thicken the potatoes.
Mix potato, egg, onions, garlic powder and paprika. If the mixture doesn't hold together well because it's too wet, add the flour a little at a time.
When the mashed potatoes are firm, form 1/2 cup each into patties with your hands. One-quarter to one-half inch thick works well. The patties will warm through as the outside cookies.
Heat the butter and olive oil together in a heavy fry pan. When the oil sizzles, place the potato pancakes in the pan. Move each pancake slightly after two minutes to keep them from sticking to the pan. Lift the edge of the pancake to check for browning. When they are brown and crispy, turn them over and fry again until brown and crispy.