Heat 2 teaspoons of the oil in a nonstick skillet over medium heat. Add the onions and saute until translucent and beginning to brown, about 4 minutes. Add the garlic and continue to cook for another 1 minute. Transfer the onion mixture to a medium bowl and allow to cool. Heat remaining 2 teaspoons of oil in the skillet and saute the mushrooms until they release all of their water, about 6 to 7 minutes. Remove from heat and cool completely. Spray an oval baking dish with cooking spray. Arrange the bread cubes in the dish. In a large bowl, beat the eggs, egg whites, milk and mustard until incorporated. Add mushrooms, onion-garlic mixture, broccoli, Parmesan and mozzarella cheeses, sun-dried tomatoes, thyme, and salt and pepper and stir to incorporate. Pour mixture over bread, making sure liquid saturates bread. Cover with plastic wrap and refrigerate overnight, or at least 8 hours. Preheat the oven to 350 degrees F. Remove the plastic wrap from strata and bake for 55 to 60 minutes, or until top forms a light brown crust.