Oven & Dish Prep: Position an oven rack in the center and preheat to 350°F (175°C). Thoroughly grease your casserole dish with butter or non-stick spray.
Sauté Aromatics & Vegetables: Place the butter in the large skillet over medium heat. Once melted, add the minced shallots and cook, stirring, for about 1 minute until fragrant. Add the diced asparagus and mushrooms to the skillet. Continue to cook, stirring frequently, until the vegetables soften and the mushrooms release their liquid, about 6-8 minutes. Remove from heat and transfer the cooked vegetable mixture to the prepared casserole dish, spreading it evenly.
Layer the Cheese: Sprinkle the shredded Swiss cheese uniformly over the layer of sautéed vegetables in the dish.
Prepare Egg Custard: In the large mixing bowl, crack the eggs. Add the milk, Dijon mustard, salt, pepper, and ground nutmeg. Whisk vigorously until the eggs are well-beaten and all ingredients are thoroughly combined.
Combine & Bake: Carefully pour the egg mixture over the cheese and vegetables in the casserole dish. Ensure the egg mixture distributes evenly. Place the dish into the preheated oven.
Bake Until Set: Bake for 40 to 45 minutes. The bake is ready when the center is set (a knife inserted near the middle should come out clean) and the top is lightly golden brown.
Rest Before Serving: Remove the bake from the oven and let it stand on a wire rack for 10 minutes before slicing and serving. This allows the eggs to fully set and makes slicing easier.