Boil pasta in salted water until 1–2 minutes shy of package directions for al dente. Drain, rinse under cold water to stop cooking, and toss with a small drizzle of olive oil.
Halve cherry tomatoes, dice cucumber and bell pepper, thinly slice red onion, and halve olives. Tear basil and set ingredients in a large mixing bowl.
Whisk together olive oil, lemon juice, Dijon, minced garlic, oregano, sugar (if using), salt, and pepper. Adjust seasoning to taste.
Combine pasta and vegetables, pour dressing over, and toss gently to coat. Fold in mozzarella and nuts last. Chill at least 30 minutes for best flavor.
Notes
Make ahead: stores well for 3–4 days in an airtight container.
Variation: add grilled chicken, shrimp, or chickpeas for protein.
Gluten-free: substitute GF pasta and follow the same method.
Adjust dressing: add a splash more olive oil or vinegar if pasta absorbs too much dressing after chilling.