Underneath monkfish's skin lies a thin, purple, iridescent membrane you have to remove before cooking. The membrane adheres snugly to the flesh and shrinks during cooking, constricting the monkish fillet and misshaping it. You have to exercise finesse when removing the membrane to preserve the flesh, and you can only slice with finesse using a sharp fillet knife. Fillet knives have thin blades made for the detail work seafood fabrication requires, and there really isn't a substitute.
Rinse the monkfish and pat it dry with paper towels. Loosen a piece of the membrane large enough to grasp between your thumb and forefinger from the wide end of the tail using a fillet knife.
Pull up on the loosened tab of membrane and position the knife blade at the point that it meets the flesh. Angle the blade of the fillet knife slightly upward.
Slide the knife back and forth gently while slowly working it towards the tapered end of the tail. The membrane will come off in a 1/2- to 3/4-inch-wide strip when you reach the end of the tail.
Loosen another tab of membrane at the wide end of the tail and slice it from the flesh. Continue pulling and slicing the membrane away in strips until the flesh is clean.
Turn your monkfish fillet 180 degrees. Start at the tail end. Slide your knife between the membrane and the fillet at the tail end of the fish. Work your knife outward until you free the membrane from the fillet. Continue to work towards the top end of the fish, sliding your knife under the membrane and cutting outward in this fashion until the membrane is only attached along the spine of the monkfish.
Cut the flap of membrane free from the monkfish's spine with your knife.