Heat oil in a large skillet.
Add onions and thyme and cook until softened, about 5 minutes.
Add garlic, mushrooms, asparagus, ½ teaspoon salt, and cook for 5 minutes, stirring as needed. Remove from heat and set aside.
Whisk together eggs, milk, mustard, parsley, pepper and ½ teaspoon of salt in a large bowl. Set aside.
Lightly grease 9 x 13 baking dish. Spread half of the bread cubes over the bottom of the dish.
Top bread with half of the vegetable mixture and sprinkle with ½ of the cheese.
Add remaining bread cubes, then layer with the rest of the vegetable mixture.
Pour egg mixture over the casserole.
Cover and chill overnight.
Preheat oven to 375°F. Bring casserole to room temperature and sprinkle cheese. over the top. Bake about 45 minutes until firm in the center and golden brown on top.