Bring a large pot of salted water to a boil. Add pasta and cook according to package directions for al dente, about 10 minutes. Drain and rinse with cold water. Transfer pasta to large mixing bowl, drizzle 1 tablespoon of olive oil, and toss well to coat.
In a large skillet, heat 1 tablespoon olive oil over medium-high heat. Add onion and cook until soft, about 2 to 3 minutes. Add carrot, summer squash, and asparagus. Cook 1 to 2 minutes. Add peas and cook for 1 more minute. Vegetables should be just cooked and still crisp. Season with salt and pepper, remove from heat, and let cool.
Whisk together red wine vinegar, lemon juice, garlic, basil, and Parmigiano-Reggiano until combined. Continue whisking and drizzle in remaining 5 tablespoons of olive oil.
To serve, toss pasta with cooked chicken, sautéed vegetables, cherry tomatoes, and Parmigiano-Reggiano dressing until well combined. Transfer to serving platter and garnish with Parmigiano-Reggiano curls.