In a large skillet, heat oil over medium-high heat. Add asparagus, mushrooms, garlic salt and 1/4 teaspoon pepper; cook and stir 3-4 minutes or until asparagus is crisp-tender.
Place half of the bread cubes in a greased 13×9-in. baking dish. Top with asparagus mixture, goat cheese and remaining bread.
In a large bowl, whisk eggs, milk, thyme, salt and remaining pepper. Pour over top; sprinkle with Parmesan cheese. Refrigerate, covered, overnight.
Preheat oven to 350°. Remove strata from refrigerator while oven heats. Bake, uncovered, 50-60 minutes or until a knife inserted near the center comes out clean. Let stand 5-10 minutes before serving. Meanwhile, prepare sauce mix according to package directions. Serve with strata.