Cook the tortellini: Bring a large pot of salted water to a boil. Add the tortellini and cook according to the package instructions, usually about 3-5 minutes. Be careful not to overcook, as you want the pasta to remain firm.
Drain and cool: Drain the cooked tortellini in a colander and rinse with cold water to stop the cooking process. Let it sit while you prepare the other ingredients.
Prepare the vegetables: While the pasta is cooling, chop the cherry tomatoes, cucumber, red bell pepper, red onion, and basil. Slice the black olives and dice the mozzarella if needed.
Make the dressing: In a small bowl or jar, whisk together the olive oil, red wine vinegar, lemon juice, Dijon mustard, minced garlic, honey, oregano, salt, and black pepper until well combined.
Assemble the salad: In a large mixing bowl, combine the cooked tortellini, chopped vegetables, black olives, mozzarella, and Parmesan cheese.
Add the dressing: Pour the dressing over the salad and toss gently to coat everything evenly.
Chill and serve: Cover the salad and refrigerate for at least 30 minutes before serving to allow the flavors to meld. Garnish with fresh basil before serving.