Preheat oven to 325°. Grease a 13x9-inch baking dish with 1 tablespoon butter; set aside.
In a large Dutch oven, melt remaining ¼ cup butter over medium heat. Add mushrooms and garlic; cook, stirring occasionally, for 12 to 14 minutes or until lightly browned; set aside. In a large bowl, toss croissants with ½ cup melted butter.
Add ham, cheese, asparagus, and mushrooms, tossing to combine.
In a large saucepan, combine milk, cream, salt, pepper, and nutmeg. Heat over medium-low heat until milk mixture is barely simmering, stirring frequently. Whisking constantly, slowly add eggs to hot milk mixture; whisk until smooth. Pour egg mixture over bread mixture; let stand for 10 minutes.
Spoon mixture into prepared baking dish. Place baking dish in a larger pan. Pour enough hot water into larger pan to reach halfway up sides of baking dish. Bake for 1 hour or until custard is set and bread is golden brown.
