Prepare Tomato Juice: Bring tomatoes and next 4 ingredients to a simmer in a large nonaluminum saucepan over medium heat; simmer 25 minutes or until mixture is soft. Cool 30 minutes. Process, in 2 batches, in a blender until smooth. Pour through a fine wire-mesh strainer into a container, discarding solids. Cover and chill 2 hours.
Prepare Cocktail: Combine kosher salt and parsley. Rub rim of a glass with lime wedge; dip rim in salt mixture to coat. Muddle tomato and basil sprig against sides of glass to release flavors. Fill glass with ice. Stir in gin and 1/2 cup Garden Tomato Juice. Sprinkle with ground pepper.