Preheat oven to 425°F. Coat a 9-by-13-inch broiler-safe baking dish with cooking spray. Place spaghetti squash halves, cut-side down, in a baking dish or on a large rimmed plate; pour 2 tablespoons water over the squash. Microwave on High, uncovered, until tender, about 12 minutes. Let stand until cool enough to handle, about 10 minutes. Scrape the squash flesh with a fork to create strands of “spaghetti”; place in a large bowl.

Meanwhile, place 1½ cups half-and-half in a medium saucepan; bring to a simmer over medium-high heat. Reduce heat to medium-low; add grated garlic. Cook, whisking occasionally, until reduced to about 1 generous cup, 8 to 10 minutes, adjusting heat and whisking as needed if starting to bubble and foam up the sides of the pan. Stir in ½ cup sun-dried tomatoes, ½ teaspoon pepper and ¼ teaspoon salt; remove from heat. Add cubed cream cheese; whisk until smooth. Gradually add ½ cup mozzarella, about ¼ cup at a time; whisk until smooth.

Add 3 cups chicken to the bowl with the spaghetti squash. Add the cheese mixture; stir and fold to combine. Pour into the prepared baking dish and top with the remaining ½ cup mozzarella, 2 tablespoons Parmigiano-Reggiano and the remaining ¼ cup sun-dried tomatoes.

Bake until the cheese has melted, about 10 minutes. Increase oven temperature to broil; broil until the cheese on top is crispy and caramelized, 1 to 2 minutes. Let stand for 10 minutes. Top with 2 tablespoons basil and the remaining ¼ teaspoon pepper.