Heat olive oil in a large skillet over medium heat. Add onions and thyme, and cook until softened, about 4 minutes. Add garlic, mushrooms, asparagus, and half of the salt and cook for 5 more minutes. Set aside. In a medium bowl, whisk together eggs, milk, mustard, parsley, pepper and the remaining salt.
Spread half of the bread cubes over the bottom of a 10x8x2-inch non-stick pan. Top with half of the vegetable mixture and a third of the crumbled goat cheese. Layer with the remaining bread cubes and vegetables, and pour the egg mixture evenly over the top. Cover strata and refrigerate overnight.
Preheat oven to 375°F Remove strata from refrigerator while the oven heats. Sprinkle remaining goat cheese over top of strata, and bake, uncovered, for 40 minutes, or until firm in the center and golden brown. Let rest 5 minutes before serving.