Grilled Shrimp Cocktail

Name: Grilled Shrimp Cocktail
Ingredients: 2 cups ketchup; 2 tablespoons freshly grated horseradish root (see Note); 2 tablespoons high-quality bottled horseradish; 2 tablespoons freshly squeezed lemon juice; 1 1/2 tablespoons Worcestershire sauce; 1 tablespoon hot sauce; 2 medium shallots, peeled, and grated on a Microplane grater; 8 cups water; 1/4 cup kosher salt; 2 tablespoons granulated sugar; 2 lemons; 4 garlic cloves, peeled, and grated on a Microplane grater; 3 cups ice; 24 (8–12-count) unpeeled shrimp (this means shrimp that come 8 to 12 to a pound); 1/3 cup canola or vegetable oil; 1/2 cup extra virgin olive oil; Juice of 2 lemons; 1/4 cup minced chives; Kosher salt; Freshly ground coarse black pepper; 2 lemons, each cut into 4 wedges for serving