In a large skillet over medium-high heat, melt butter. Add mushrooms and onions, cooking for 8-10 minutes until mushrooms release their liquid and turn golden brown. Add garlic and cook 1 minute more. Remove from heat and let cool slightly.
Grease a 9x13-inch baking dish. Spread bread cubes evenly in the dish.
Layer the mushroom mixture over the bread, then sprinkle with 1½ cups of Swiss cheese. Reserve remaining ½ cup for topping.
In a large bowl, whisk together eggs, milk, thyme, salt, pepper, and nutmeg until well blended.
Pour egg mixture evenly over the casserole, pressing down gently to ensure all bread is moistened. Cover and refrigerate overnight or at least 4 hours.
To freeze: Wrap tightly with plastic wrap followed by aluminum foil. Freeze for up to 2 months. Thaw in refrigerator overnight before baking.
When ready to bake, preheat oven to 350°F and let casserole sit at room temperature for 20 minutes.
Cover with foil and bake for 35 minutes. Remove foil, top with remaining ½ cup Swiss cheese, and bake uncovered for 20 minutes more until puffy and golden.
Let stand 10 minutes before serving. Garnish with fresh parsley and enjoy!