Preheat oven to 375°F. Place squash cut side up on a foil-lined, rimmed baking sheet. Add 1/4 inch water to sheet. Bake until edges are golden brown and squash is easily pierced with the tip of a sharp knife, about 35 minutes. Set aside until just cool enough to handle.
In a large nonstick skillet, heat oil on medium. Add chicken and cook for 2 minutes. Flip chicken and add pear and sage. Sauté, stirring occasionally, until chicken is cooked through, about 5 minutes. Transfer chicken, pears and pan drippings to a large bowl.
With 2 forks, scrape stringy squash flesh from skin, separating into strands. Add squash flesh to bowl with chicken mixture. Add chives and toss to combine. To serve, top with cheese.