Prepare a small shallow bowl or plate with about ½ cup flour for dredging. In a large bowl, combine the mashed potato, ¼ cup flour, and salt and other add-ins, if using. Mix well.
Use your hands to scoop out about ⅓ cup of the potato mixture and form round patties with your hands. Add more flour to the mixture, if they are falling apart (see notes).
Dredge each patty on both sides in the plate of flour. Gently shake off excess flour.
Preheat a skillet with olive oil (~½ - 1 tablespoon olive oil, depending on the size of the skillet). Carefully place the dredged patties into the hot oil and fry undisturbed over medium heat for 3-5 minutes per side, or until golden brown. Place on a paper towel to absorb the excess oil. Repeat until all the patties are cooked. Allow to cool for 5 minutes and serve warm (see ideas for serving).
You might need to add more flour if the patties are falling apart while dredging or cooking/flipping. The amount of flour will depend on how wet/dry your mashed potatoes are.
Smaller patties are easier to fry without them breaking.
Do a frying test by frying 1 patty and making sure it holds together. If yes, the mixture is just right. If it falls apart, you might need more flour.
Use a non-stick skillet to prevent the crispy crust from sticking to the skillet.
Shredded cheese (any kind, ¼ - 1 cup, to taste)
Fresh parsley, chives, scallion, or dill, finely diced (1 tablespoon - ¼ cup, to taste)
Garlic powder (~½ - 1 teaspoon, to taste)
Smoked sweet paprika (~1-2 teaspoons, to taste)
Cooked crumbled bacon (~4-6 slices)
Finely diced cooked ham (~½ - 1 cup)
Seasoned breadcrumbs (instead of the flour for the dough and for dredging)