Preheat oven to 400°F. Line a large rimmed baking sheet with parchment paper. Brush cut sides of squash halves with 1 tablespoon olive oil. Arrange the squash halves cut-side down on the prepared baking sheet. Roast until tender when pierced with a fork, 35 to 40 minutes.
Meanwhile, stir 2 cups chicken, ⅔ cup yogurt, ⅓ cup chopped sun-dried tomatoes, ¼ cup pesto, ½ teaspoon garlic powder, ¼ teaspoon plus ⅛ teaspoon salt and ¼ teaspoon crushed red pepper together in a medium bowl until combined; set aside, uncovered, at room temperature, until ready to use.
Flip the squash halves. Divide the chicken mixture between the squash halves; top with ½ cup goat cheese. Drizzle with 2 tablespoons sun-dried tomato oil.
Roast until the cheese is lightly browned and the filling is hot throughout, about 15 minutes.
Using a fork and tongs, scrape the chicken mixture and the long strands of squash flesh into a medium bowl. Add the remaining 1 tablespoon sun-dried tomato oil; toss to combine. Return the mixture to the squash shells; cut each in half crosswise. Transfer each piece to a plate; drizzle with 2 tablespoons balsamic glaze. Sprinkle with ¼ cup basil.
