Prepare the Spaghetti Squash
Preheat the oven to 425°F. Coat a 9-by-13-inch broiler-safe baking dish with cooking spray.
Place the spaghetti squash halves cut-side down in a microwave-safe dish or on a large rimmed plate. Add 2 tablespoons of water.
Microwave on High, uncovered, for 12 minutes, or until the squash is tender. Let it cool for 10 minutes.
Use a fork to scrape the squash flesh into spaghetti-like strands and transfer them to a large bowl.
Make the Sauce
Heat 1½ cups of half-and-half in a medium saucepan over medium-high heat until it begins to simmer. Reduce the heat to medium-low.
Add the grated garlic and cook, whisking occasionally, for 8–10 minutes until the liquid reduces to about 1 cup.
Stir in ½ cup of the chopped sun-dried tomatoes, ½ teaspoon of pepper, and ¼ teaspoon of salt. Remove the saucepan from heat.
Gradually whisk in the cream cheese until smooth, then whisk in ½ cup of shredded mozzarella, a little at a time, until fully melted.
Assemble the Casserole
In the large bowl with the spaghetti squash strands, add 3 cups of shredded cooked chicken. Pour the creamy sauce over the mixture and stir to combine.
Transfer the mixture to the prepared baking dish.
Add the Toppings
Sprinkle the remaining ½ cup mozzarella cheese evenly over the top.
Add 2 tablespoons of grated Parmigiano-Reggiano and the remaining ¼ cup of sun-dried tomatoes.
Bake and Broil
Bake the casserole for 10 minutes, or until the cheese is melted and bubbling.
Switch the oven to broil and broil for 1–2 minutes, or until the cheese is crispy and caramelized.
Garnish and Serve
Let the casserole rest for 10 minutes before serving.
Garnish with fresh basil leaves and sprinkle with the remaining ¼ teaspoon of pepper.