Make the dressing: Add the oil, vinegar, lemon juice, garlic, oregano, salt, black pepper, and red pepper flakes to a small bowl or jar. Whisk or shake vigorously for about 30 seconds to combine. Set aside.
Add the cooked tortellini, arugula, chickpeas, cherry tomatoes, cucumber, olives, basil, and mozzarella cheese to a large bowl. Pour the dressing on top and toss to evenly coat.
Chill for at least 20 minutes, or until serving.
