Cook the Orzo: Bring a medium pot of salted water to a boil. Add the orzo and cook according to package directions until al dente (about 8 to 9 minutes).Drain well in a fine-mesh colander.
Whisk the Dressing: While the pasta boils, combine the extra virgin olive oil, red wine vinegar, fresh lemon juice, Dijon mustard, minced garlic, dried oregano, salt, and black pepper in a small bowl or liquid measuring cup. Whisk vigorously until emulsified.
Dress the Warm Pasta: Immediately transfer the warm, drained orzo to a large mixing bowl. Pour half of the prepared dressing over the warm pasta and toss well to coat. This prevents the orzo from clumping as it cools.
Combine the Salad: Once the orzo has cooled slightly, add the rinsed chickpeas, rinsed capers, halved tomatoes, diced cucumber, red onion, kalamata olives, and chopped parsley to the bowl.
Toss and Finish: Pour the remaining dressing over the salad and gently toss to combine. Fold in the crumbled feta cheese just before serving, and finish with fresh chopped basil or mint and a squeeze of lemon to brighten. Serve at room temperature or chilled.