Preheat the oven to 400°F (200°C) so it is fully heated before the casserole is placed inside.
In a large mixing bowl, place the cooked and cooled spaghetti squash, eggs, shredded mozzarella, and Greek yogurt. Stir until the ingredients are evenly incorporated and the mixture has a cohesive, creamy texture.
Add salt, pepper, minced garlic, and dried thyme to the squash mixture. Mix thoroughly so the seasonings are distributed throughout the casserole base.
Transfer the seasoned mixture into a lightly greased casserole dish. Spread it into an even layer, then sprinkle the grated Parmesan cheese over the top to create a savory crust as it bakes.
Bake the casserole in the preheated oven for about 30 minutes, or until the surface is lightly browned and the mixture is set.
Remove the casserole from the oven and let it rest for 10 minutes before serving. This allows the dish to firm slightly, making it easier to portion.