1 tablespoon extra virgin olive oil
Heat oil in a large skillet over medium heat. Add onions and thyme and cook until softened, about 4 minutes. Add garlic, mushrooms, asparagus and 1/2 teaspoon of the salt and cook 5 more minutes longer. Set aside. Meanwhile, whisk together eggs, grated cheese, milk, mustard, parsley, pepper and remaining 1/2 teaspoon salt in a large bowl. Set aside. Spread half of the bread over the bottom of a lightly greased 9-x13-inch baking dish. Top with half of the mushroom mixture and a third of the goat cheese. Layer with remaining bread and mushroom mixture, and pour egg mixture evenly over the top. Cover strata and chill overnight. Preheat oven to 375°F. Remove strata from refrigerator and bring to room temperature while the oven heats. Sprinkle remaining two-thirds cheese over the top and bake until firm in the center and golden brown on top, about 45 minutes. Let rest 10 minutes before serving.