Preheat your oven to 375°F (190°C) and lightly grease a baking dish with butter.
In a mixing bowl, combine shredded cooked chicken, diced squash, chopped kale, minced garlic, and chopped sage. Toss everything together to evenly distribute the ingredients.
Drizzle the olive oil over the mixture, then add salt and black pepper. Mix again until all ingredients are coated and well combined.
Transfer the mixture into the prepared baking dish, spreading it out into an even layer. Sprinkle shredded cheese over the top, covering everything generously.
Place small pats of butter on top of the cheese, then put the casserole into the oven. Bake for about 30 to 35 minutes, or until bubbling and golden around the edges.
Once baked, remove from the oven and let sit for a few minutes. The casserole will set slightly and develop a crispy top layer.
Slice into portions and serve hot, enjoying the hearty, cheesy, seasonal flavors with a rustic appearance.