If using fresh pumpkin, peel and chop it into small cubes. If using canned pumpkin, simply measure it out and set aside.
In a medium skillet, heat the olive oil over medium heat. Add the chopped green garlic and sauté for about 3-4 minutes, just until it becomes fragrant and softened.
In a large mixing bowl, combine the sautéed green garlic, chopped pumpkin, heavy cream, shredded mozzarella cheese, grated Parmesan cheese, salt, pepper, and thyme. Stir well until all ingredients are well mixed.
Pour the pumpkin and green garlic mixture into a greased 9x13 inch baking dish, spreading it evenly.
Break the garlic toast into bite-sized pieces and sprinkle evenly over the top of the pumpkin mixture.
Place the baking dish in the preheated oven and bake for 30-35 minutes, or until the pumpkin is tender and the garlic toast is golden brown and crispy.
Once done, remove from the oven and let it cool for a few minutes. Garnish with fresh parsley before serving.
Serve warm as a comforting side dish or a vegetarian main course.