Grease a 13-by-9-inch baking dish with unsalted butter. Set aside.
In a large nonstick pan set over medium heat, cook the breakfast sausage, breaking it apart into crumbles. Remove it to a paper towel-lined plate and drain any drippings from the pan.
Add 1 tablespoon of unsalted butter to the pan. Once the butter has melted, add the diced onion and cook, stirring occasionally, until it's soft and translucent, 3 to 5 minutes.
Transfer the onion to a medium bowl then stir in the cooked sausage.
In a large bowl, whisk together the eggs, milk, Dijon mustard, salt and pepper. Stir in the cheese and asparagus.
Scatter the bread cubes evenly in the greased baking dish, then top the bread with the sausage mixture and asparagus. Pour the egg mixture over the layers of bread and sausage. Cover the dish securely with plastic wrap and refrigerate it overnight, or for about 8 hours.
Remove the strata from the fridge, discard the plastic wrap, and let it rest at room temperature for 30 minutes.
Preheat the oven to 350ºF. Place the strata on a baking sheet and bake it, uncovered, for 35 to 45 minutes, until the eggs are fully cooked. Let the strata rest for 5 minutes before slicing and serving.