Heat the oil in a small saucepan until hot but not smoking. Drop the plantain rounds into the hot oil 3 or 4 at a time and cook until lightly browned, 2 to 3 minutes. Remove and drain on paper towels.
Set each fried plantain on a flat side, and using a rolling pin, frying pan, or whatever else you think will work, squash it as flat as you can. It should have a circular shape.
Return the flattened plantain sections to the hot oil 3 or 4 at a time and cook until the entire surface is golden brown, about 2 minutes. Remove the plantains from the oil, drain, and season liberally with salt and pepper. Serve hot or at room temperature.