Preheat the oven to 375°F (190°C).

Prepare the butternut squash by peeling, seeding, and cutting it into 1-inch cubes.

In a saucepan, cook quinoa with vegetable broth according to package instructions.

In a large skillet, heat olive oil over medium heat, then add onions and garlic, sautéing until translucent.

Add the cubed butternut squash to the skillet, along with cinnamon, nutmeg, salt, and pepper; cook for about 10 minutes until slightly tender.

Combine the cooked quinoa with the sautéed mixture in a large bowl, mixing well.

Transfer the mixture into a greased baking dish, and sprinkle the shredded cheese on top.

Bake in the preheated oven for 30-35 minutes, until the cheese is bubbly and golden brown.

Let the casserole rest for a few minutes before serving to allow it to set slightly.