Bring a large pot of salted water to a boil over high heat.

Add the pasta and cook according to package directions until al dente, then drain in a colander and rinse under cool water to stop the cooking.

Spread the pasta in the colander or a tray to cool completely while you prepare the other ingredients.

In a small bowl, whisk together extra-virgin olive oil, balsamic vinegar, red wine vinegar, Dijon mustard, dried oregano, kosher salt, black pepper, and red pepper flakes until emulsified.

In a large mixing bowl, combine cherry tomatoes, fresh mozzarella balls, red onion, minced garlic, and sliced basil.

Add the cooled pasta to the large bowl with the tomato mixture and toss gently to combine.

Pour the dressing over the pasta salad and toss until everything is evenly coated.

Cover the bowl and let the salad rest in the refrigerator for at least 20 minutes to allow the flavors to meld.

While the salad chills, heat 1 tablespoon olive oil in a skillet over medium heat.

Add the torn crusty bread pieces to the skillet, season with a pinch of kosher salt, and toast, stirring often, until golden and crisp, then let cool.

Just before serving, sprinkle the toasted bread pieces and grated Parmesan cheese over the salad and toss lightly.

Taste and adjust seasoning with additional salt, pepper, or vinegar if needed before serving.