Position one oven rack in the middle and one near the top of the oven. Preheat the oven to 375°F. Grease a 9×13-inch baking dish with butter.
Arrange the torn croissants in a single layer in the prepared baking dish. Add the ham and 1 cup of Gruyère cheese, evenly distributing both over the croissants.
In a medium bowl, whisk together the melted butter, eggs, milk, yogurt, salt, pepper, garlic powder, Dijon, and fresh thyme until well combined.
Carefully pour the egg mixture into the baking dish, ensuring all croissant pieces are evenly coated. Sprinkle evenly with the remaining 1/2 cup Gruyère and Parmesan cheese.
Transfer to the oven on the middle rack and bake for 30 minutes. After 30 minutes, move the casserole to the higher rack and bake for an additional 15 to 20 minutes, or until the top is a deep, golden brown and the center is set.
Allow to cool for 10 to 15 minutes. Garnish with the remaining thyme and cut into 8 squares. Enjoy!