1. Cut through the skin around the back of the head with a small, sharp knife.
2. Using fish pliers, pull away about 1 inch (2.5 cm) of skin from all the way around the head.
3. Hang the eel up by the head, with a meat hook or a piece of string, from something very secure and with plenty of room in which to work. Using a second pair of pliers, grab hold of some skin on either side of the eel and start to pull it away.
4. As soon as the skin starts to come away more cleanly, firmly and steadily pull it down toward the tail. As you near the tail it will start to get a little harder, but just give it a vigorous final tug and it will come away completely, over the tail.
5. To fillet the eel, lay it on a chopping board and cut off the head. Using a sharp, thin-bladed, flexible knife, make a shallow cut along the backbone of the fish, just above the line of bones. Start to cut away the fillet, keeping the blade of the knife as close to the bones as you can.
6. As soon as you can get the whole blade of the knife under the fillet, rest your other hand on top of the fish and cut the fillet away in one clean sweep, down toward the tail. Turn the eel over and repeat on the other side.