In a large bowl, stir together the mashed potatoes, flour, cheese, and eggs.
If you're using leftover mashed potatoes that had a lot of milk in them, you will likely need to add more flour until the mixture is thick and workable. You should be able to take a ball of dough in your hand and press it into a patty without it sticking to your hands. Add flour until it reaches this consistency. Alternately, if the dough is too thick and dry, add milk one tablespoon at a time until it reaches the right consistency.
Once your dough is the right consistency, stir in the ham, onion, garlic, salt, and pepper.
Heat a large skillet over medium heat. Add just enough oil to lightly cover the bottom of the skillet.
When the oil is hot, take a golf ball sized ball of dough and press into a patty. Place in the skillet. Repeat with the rest of the dough, being sure not to crowd the pan.
Cook for 2 minutes per side or until golden brown.
Drain on a paper towel lined sheet.
Sprinkle with extra cheese and chives before serving.