Combine the potatoes, flour (starting with ⅓ cup), salt, and pepper in a small mixing bowl. Lightly mix with a spoon to combine.
Make a well in the center of the mixture, crack an egg into the well, and whisk, then fold it into the potato mixture.
Gently stir in the chives.
Add more flour to the mixture as needed, so that it can be easily formed into patties. Be careful not to use too much flour.
Melt the butter in a skillet over medium heat.
Form 4 patties with the potato mixture and place them into the skillet, cooking the patties until golden brown on each side, about 5 to 7 minutes per side.