Cook pasta according to package directions. Drain reserving 1/4 cup of the cooking water; set cooking water aside. Return pasta to pan; keep warm.
Meanwhile, place squash cut sides down in a microwave-safe baking dish with 2 tablespoons water. Microwave, covered, on 100% power (high) for 7 to 10 minutes; rearranging once halfway through cooking. Let stand, covered, for 5 minutes.
Sprinkle chicken with 1/4 teaspoon of the salt and 1/4 teaspoon of the pepper. In large skillet heat oil over medium-high heat. Add chicken; cook for 6 to 8 minutes or until golden brown and no longer pink in the center. Remove chicken from skillet; keep warm.
In a small bowl whisk together milk and flour. Whisk in chicken broth and reserved cooking water; set aside. Scrape flesh from squash into a medium bowl; coarsely mash. Transfer 1 1/4 cups squash and the remaining salt to hot skillet. Stir in broth mixture, the 2 tablespoons Parmesan cheese, and the 1 tablespoon sage. Bring to boiling; cook and stir for 1 minute more. Pour sauce mixture over pasta in pan. Add chicken; toss to combine.
To serve, divide mixture among 5 serving plates. Sprinkle evenly with 1/3 cup Parmesan cheese, 2 tablespoons snipped fresh sage, and the remaining 1/4 teaspoon pepper.