Combine the 1 cup room temperature water, the salt, and sugar in a large bowl and stir to dissolve. Add the ice and set aside.
Fill a medium bowl with cold water. Use a pair of kitchen shears to cut the shrimp open across the back, from front to tail, exposing the inner vein. Submerge in the water and remove the vein, which should float off and sink. Avoid removing the shell if possible. As each shrimp is deveined, transfer it to the cold brine. Once they're all in, leave to brine for another 30 minutes.
When you've got about 10 minutes remaining, bring an inch of water to a boil in a saucepan large enough to accommodate your steamer basket, over high heat.
Drain the shrimp and briefly rinse under cold water. Arrange on your steamer basket, minimizing overlap. (You may need to do this in a couple of batches.) Drop the heat to medium-high, cover, and steam for 3 minutes.
Meanwhile, rinse out the bowl and add 2 to 3 cups of ice to it.
Transfer the shrimp to the ice bowl and toss a few times to stop the cooking process. Toss every couple of minutes until the shrimp have chilled through, then either serve or refrigerate in airtight containment for up to 24 hours. Alternately, place in the freezer, tossing every few minutes for 15 to 20 minutes or until thoroughly chilled.
Peel the shrimp and serve with the cocktail sauce. Or serve with the shells on and let your guests do the peeling. That'll slow them down a little.