Preheat the oven to 375 degrees. With a sharp knife, prick the squash in 2 to 3 places and set it on a baking pan. Bake for 30 minutes. Remove it from the oven, cut it in half lengthwise and scoop out the seeds. Return the squash halves, cut side down, to the baking sheet and bake for another 30 minutes or until tender.
While the squash is cooking, heat the butter in a medium sized saucepan. Add the flour and cook for 1 minute, stirring continuously. Add the garlic, broth and bring the mixture to a boil, whisking all the while. Add the chicken and simmer, uncovered, for 3 to 5 minutes or until chicken is cooked through. Remove from the heat and stir in the milk or heavy cream. Season with 1 teaspoon of salt and pepper to taste; set aside and reheat over low heat when you remove the squash from the oven.
Remove the squash from the oven. With a fork rake the squash until it separates fully into strands and only the shell remains. Transfer the strands to pasta bowls. Remove the chicken sauce from the heat, stir in the cheese and parsley or basil and adjust the seasoning. Serve the sauce over the strands of squash.