Monkfish is one of the most fearsome-looking sea creatures, but its flesh is unusually sweet and fine. Bottom dwellers with very large heads and relatively small bodies, they burrow themselves into the ocean floor and wait for prey to swim past -- at which point they launch themselves from under cover and trap the meal in their massive jaws. Monkfish are available at most good fishmongers, and their uncomplicated bone structure makes them easy to fillet.
Place the monkfish on your cutting board. If the fish is whole, begin by removing the oversized head. Grasp the head by the eye sockets with your non-dominant hand and cut diagonally, backward from the corner of the jaw to the backbone. Repeat from the other side, then sever the backbone. Set the head aside.
Loosen the skin around the cut end of the monkfish tail, using the tip of your boning knife. Once you've freed enough skin to grip, hold down the monkfish with one hand and pull away the skin with your other. It should turn inside out and peel away, like taking off a sock.
Sever the tail and skin from the monkfish, leaving the flesh behind. Cut the fillets away from the backbone by slicing straight down vertically, or turning the tail on its side and slicing horizontally. Either technique works.
Set aside the tail for making fish stock, if desired. Trim the pale pink-purple membranes away from the fillets by lifting them from the flesh and cutting with the knife.
Cut the fillets into portions as desired. Cook immediately, or refrigerate until needed.
If your monkfish came with the head attached, the large cheek muscles can be cut away from the sides of the head and skinned. On a large monkfish, this can yield two more portions of usable flesh. If your fishmonger buys monkfish whole and cuts off the heads, you might be able to purchase the cheek muscles separately at a good price.
Clean and sanitise any surfaces that have come into contact with the uncooked monkfish.