For the croutons: Preheat the oven to 275 degrees F.
Cut the bread into 1-inch cubes and arrange on a baking sheet. Drizzle the cubes with the olive oil and toss to coat. Put the pan in the oven to slightly crisp ("stale") the bread without toasting it, 20 to 25 minutes. Remove and allow to cool.
For the dressing: In a small jar, shake together the olive oil, vinegar, salt and pepper.
For the salad: In a large bowl, combine the tomatoes, cucumbers, onions and bread cubes. Pour the dressing over the salad ingredients, tossing gently. Add the sliced basil and Parmesan shavings and toss again. Sprinkle with more salt and pepper and serve garnished with whole basil leaves.
If you want to avoid using the oven, you can cut up the bread and let it dry out at room temperature overnight.