Boil a small pot of water; cook basil leaves in water for 15 seconds then remove with a slotted spoon and dunk in an ice water bath; drain and dry. Using a food processor or blender, puree basil and broth until smooth. Let mixture sit for 2 hours at room temperature. Strain mixture through a fine chinois into a bowl, disposing of solids. Whisk salt and vinegar into mixture.
Lay tomato slices on a platter and top with basil sauce, cheese, pepper, and fresh basil.