Step 1 Combine salt and water in a large mixing bowl and stir to dissolve the salt. Add ice. Set the brine aside while you prepare the fish.
Step 2 Cut the fish into 2 1/2-inch wide by 1-inch thick strips along the length of the fillet. (After brining, you'll cut the fish into cooking portions). Be sure to use the sharpest knife in the drawer for this and work slowly and carefully. Remove the blood line from the skin side of the fillets. Place the fillets in the brine and leave for 1 hour.
Step 3Remove the fish from the brine, dip briefly in salt-free ice water and dry it thoroughly with paper towels. Lay the fillets out flat on parchment-lined baking sheets and refrigerate, unwrapped, for a minimum of 4 hours.
Step 4Cut the fillets into 5- to 6-ounce portions, and prepare as you wish.
Step 5To grill the halibut, heat the grill over medium-high heat until hot and clean it well with a wire brush. Brush both sides of the fish with a very thin coating of mayonnaise and sprinkle with salt, if desired.
Step 6Just before putting the fish on the grill, wipe down the grate with an old rag that has been briefly dipped in cooking oil. Place the fish on the grill and, after a minute or so, turn the fillets at a 45-degree angle to mark them. After another minute, flip the fillets. Cook for 2 more minutes. The flesh should yield to gentle pressure when pressed (you can also use a cake tester to test the doneness of the fillet; when the fish is properly cooked, a cake tester will pass through with only gentle resistance).
Step 7Remove the fish to rest, preferably on a rack, for 5 minutes before serving.