Preheat the oven to 400 degrees F. Cut the squash in half lengthwise and remove the seeds and any visible strings. Place the two halves cut-side down on a baking sheet and bake until soft, about 40-45 minutes.
In the meantime, coat a 13-by-9-inch baking dish with cooking spray. Heat the oil in a large saucepan over medium heat. Add the onion and saute until translucent, 5 to 7 minutes. Add the garlic and cook until fragrant, 2 minutes. Add the kale and chicken and cook down, about 5 minutes. Turn off the heat.
In a large bowl, combine the ricotta, egg, 1 ½ cups mozzarella cheese and the cooked kale mixture. Stir until well combined. If you don't want to dirty another bowl, you can simply add to the pot. Just be sure to wait until the kale mixture cools down a bit.
Spread ½ cup of tomato sauce on the bottom of the prepared baking dish, then spread half of the cooked spaghetti squash on top. Top with half of the ricotta mixture followed by 1 cup of tomato sauce. Repeat the layers once more, ending with a layer of sauce; sprinkle the top with the remaining ½ cup of mozzarella cheese.
Bake until it is hot and the cheese is melted, bubbling and slightly browned around the edges, 25 to 30 minutes. Let stand for about 10 minutes, then top with basil, if using, and serve.