Step 1In a bowl, whisk together the ketchup, Worcestershire sauce, Tabasco, horseradish and red wine vinegar. Taste and adjust flavorings if desired. This makes a generous 1 ½ cups cocktail sauce, which will keep, covered and refrigerated, about 1 week.
Step 1In a large, nonreactive bowl, whisk together the water, salt, bay leaf, garlic, red pepper, vinegar, peppercorns, halved lemons and their juice and sugar until the salt and sugar are dissolved. Add the shrimp and weight down with a plate to keep the shrimp submerged. Refrigerate for 1 hour to brine the shrimp.
Step 2Drain the shrimp, discarding the brine. Heat a grill over medium-high heat until hot.
Step 3Lightly oil the grill grates and grill the shrimp until they are firm and pink and the flesh is opaque, 2 to 3 minutes on each side. Remove the cooked shrimp and toss with the butter and chives. Serve the shrimp warm with a bowl on the side for collecting the discarded shell pieces.