First, place your knife diagonally across the fish, just behind the head, collar and pectoral fins. Slice into the fish until you reach the backbone. Turn the fish so the belly is towards you. Place your knife at the head end, just above the backbone. Angle your knife blade slightly up (away from the backbone), and run the knife along it, pulling the meat up as you cut between the fillet and the bone. Then with your knife perpendicular to the fish, finish slicing the fillet from head to tail, using the backbone as a guide and stopping about 1-inch short of the tail. Flip the fish over and do the same on the other side.
Next, cut the spine of the fish at the base of the tail. This will separate the tail and the two attached fillets from the rest of the fish.
Using the heel of your knife, make a cut through the backbone just behind the head and the collar. The head should be able to sit on a flat surface with the mouth pointed up at an angle and each side of the pelvic fin portion of the fish flared to each side, supporting the head. The meat on the pelvic fin area can be a delicacy or fishy depending on the fish, so this is primarily for presentation. You may need to use your cleaver to flatten it slightly, so it sits at the correct angle. Set aside.
Back to the fillets. With your knife at a slight angle, carefully score the fillets lengthwise—deep enough to reach the skin beneath without actually cutting through the skin. (Slicing the fish at an angle makes each segment longer with more surface area, and also helps prevent the knife from cutting through the skin.)
Space the cuts ½ inch apart (you should be able to make 3-4 cuts easily, depending on how fat your fish is). Next, score the fish crosswise (again, at an angle), cutting down and away from the tail. Once again, make the cuts deep enough to reach the skin without cutting through it. This will create a crosshatch pattern. Each fillet segment should be roughly a ½-inch square.
Dip the fish into an ice bath to rinse off any debris from processing and to tighten up the meat. Pat thoroughly dry with a paper towel. You should have two pieces: the head with the pectoral fin pieces attached and one large tail piece with the two cross-hatched fillets attached to it.