Roughly chop the tomatoes into 3/4-inch pieces and put them in a non-metallic bowl. Sprinkle with sea salt (if desired) and allow them to sit for an hour.
Tear the bread into rough bite-size chunks.
Squash the tomatoes with your hands, discard liquid and then toss with bread, onion, basil, vinegar and olive oil. Season with pepper and salt (if desired) to taste. Garnish top with extra basil leaves.
Serve with generous portion of parmesan cheese shavings over top. Best served immediately.