Step 1: In a large mixing bowl, combine the cold mashed potatoes, shredded cheddar cheese, chopped chives, beaten egg, 1/2 cup flour, garlic powder, and black pepper.
Step 2: Stir the mixture until all ingredients are well incorporated. If the mixture is too sticky to handle, add an additional 1-2 tablespoons of flour.
Step 3: Scoop about 1/3 cup of the potato mixture and shape it into a ball, then flatten it into a patty about 1/2 inch thick.
Step 4: Place a small amount of flour on a plate and lightly dredge each side of the patties, shaking off any excess.
Step 5: Heat the vegetable oil in a large non-stick or cast-iron skillet over medium-high heat until shimmering.
Step 6: Carefully place 3-4 patties into the hot skillet, ensuring you do not overcrowd the pan.
Step 7: Fry the patties for 3-4 minutes on the first side until they are deep golden brown and crispy.
Step 8: Carefully flip the patties using a thin spatula and cook for another 3-4 minutes on the other side.
Step 9: Transfer the cooked patties to a paper-towel-lined plate to drain any excess oil.
Step 10: Serve the fried mashed potato patties immediately while hot, topped with a dollop of sour cream and extra chives.