With a sharp knife, carefully cut about four 1/2-inch slits in the squash to allow steam to escape while it cooks. Put the squash on a baking sheet and roast for about 1 1/2 hours. Let the squash cool for about 20 minutes on the baking sheet.
Meanwhile, bring a pot of water to a bowl and prepare an ice water bath. Blanch the broccoli briefly in the boiling water, then transfer to the ice bath to cool. Drain and set aside.
Cut the squash in half lengthwise and use a spoon to scoop out the seeds. With a large fork, pull the squash fibers away from the outer peel (use a towel to hold the squash in place if necessary). Reserve the squash and the outer peel.
Preheat the broiler.
Put the milk and garlic in a saucepan over medium-high heat and heat until it just begins to simmer, about 3 minutes. Reduce the heat to medium, add the Neufchatel cheese, Parmesan and 1 cup of the mozzarella and stir until melted. Turn off the heat, stir in the yogurt and season with salt and pepper. Remove the garlic. Add the squash, chicken and broccoli and toss to coat. Divide the mixture between the reserved squash peels. Top with the remaining 1/2 cup mozzarella.
Place the stuffed squash onto a baking sheet and broil until the cheese is melted and bubbly, a few minutes.