Start with chilled mashed potatoes straight from the fridge; this makes them firmer and easier to work with.

In a large bowl, combine the mashed potatoes, egg, flour, grated cheddar (if using), chives, salt, pepper, and a pinch of paprika. Mix well until evenly combined. If the mixture feels too wet, add a tablespoon of flour at a time until it can be formed into patties.

Divide the mixture into 8 even portions for small cakes or 4 for larger patties. Shape each portion into a compact patty about 1/2 inch thick. Place formed patties on a tray and refrigerate for 10 minutes to firm up.

Heat a large skillet over medium heat and add the butter and oil. Allow the butter to melt and the oil to shimmer but not smoke.

Carefully place the chilled patties in the skillet, leaving space between them. Fry without moving them for 4–5 minutes, or until the bottoms are deep golden brown and crisp.

Flip the patties gently with a spatula and fry the second side for another 3–4 minutes, pressing lightly to ensure even contact with the pan. Adjust heat as needed to avoid burning.

Transfer cooked patties to a paper towel-lined plate to drain any excess oil. Keep warm in a low oven (about 200°F / 95°C) while you finish the rest.

Serve hot with sour cream, applesauce, or a simple green salad. Enjoy immediately for the best contrast between crispy exterior and creamy center.