Unlike many fish, eel should be skinned before being filleted because its tough skin is difficult to cut from the filleted flesh.
Lay the skinned eel flat on its side on a cutting board. With a sharp knife, cut across the flesh to the backbone just below the head. Alternatively, cut off the head and discard it.
Starting at the head, run a flexible knife along the center of the back and cut away the fillet. Repeat on the other side of the backbone to remove the second eel fillet in the same way.