Core and halve tomatoes, then cut each half into 4-5 wedges. Toss wedges with salt in a large bowl; let rest until small pool of liquid accumulates, 15-20 minutes.
Meanwhile, whisk oil, vinegar, garlic, basil, anchovies, and pepper to taste in small bowl. Pour mixture over tomatoes and accumulated liquid; toss to coat. Set aside to blend flavors, about 5 minutes.
Add bread cubes; toss to combine. Adjust seasonings. Sprinkle with some freshly grated Parmesan cheese and serve immediately.