In a medium bowl, stir together lemon zest and juice, mayonnaise, ¼ teaspoon salt, parsley, tarragon, horseradish, mustard, and cayenne until combined. Cover and refrigerate until ready to use.
In a large Dutch oven, heat 8 cups water, lemon slices, bay leaves, red pepper, and remaining ¼ cup salt over medium-high heat until simmering. Add shrimp; cook until bright pink and firm, about 4 minutes. Remove using a slotted spoon, and place in an ice water bath until cool. Drain well. Peel shrimp.
In a large bowl, stir together shrimp and rémoulade sauce until coated. Serve over arugula. Garnish with tarragon, if desired.
Recipe by Louisiana Cookin' at https://louisianacookin.com/classic-shrimp-remoulade/