Combine the salt, sugar, and water in a mixing bowl, stirring to dissolve. Add the ice and set aside while prepping the shrimp.
Use a pair of scissors to remove the veins from the shrimp without removing the shell. Put the shrimp in the brine and refrigerate for 20 to 25 minutes.
Meanwhile, make the cocktail sauce: Take the tomatoes, chili sauce, horseradish, sugar, pepper, and salt for a spin in a food processor until almost smooth. Refrigerate.
Place a half-sheet pan or foil-lined broiler pan about 8 inches under the broiler and heat on high for 5 minutes.
Remove the shrimp from the brine and rinse under cold water. Dry thoroughly on paper towels. In the same bowl (discard the brine), toss the shrimp with the oil and sprinkle with the Old Bay. Arrange the shrimp on the sizzling-hot sheet pan and slide back under the broiler. Set your timer for 2 minutes.
Clean out the bowl and stick it in the freezer
After 2 minutes, flip the shrimp quickly with tongs and return to the broiler for 1 minute.
Transfer the shrimp to the chilled bowl and toss a few times to knock down the heat. Place in the freezer, tossing every few minutes until the shrimp are thoroughly cooled (this shouldn’t take more than 15 minutes).
Peel the shrimp and serve alongside the sauce.