Preheat oven to 375°F (190°C).
Season chicken breasts with salt and pepper. Heat oil in a pan over medium heat. Cook chicken for 6-8 minutes per side, or until internal temperature reaches 165°F (74°C). Shred and set aside.
Boil squash cubes for 10-12 minutes, or until easily pierced with a fork. Drain.
In a large bowl, mix squash, shredded chicken, onion, and garlic.
Add broth, cream, ¾ cup cheese, thyme, ½ tsp salt, and ¼ tsp pepper. Stir well.
Transfer mixture to a 9x13 inch baking dish.
Sprinkle remaining cheese and breadcrumbs on top.
Bake for 25-30 minutes. Casserole is done when edges are bubbling and top is golden brown.
Let cool for 5 minutes before serving.
Serve hot.
