Heat 2 teaspoons of the oil in a medium non-stick skillet over medium-high heat. Add the onion and cook, stirring, until softened and beginning to brown, 3-5 minutes. Add the garlic and continue to cook for 1 minute. Transfer the onion mixture to a medium bowl and let cool.
Heat the remaining 2 teaspoons olive oil in the same skillet over medium-high heat and cook the mushrooms; stirring a few times, until their water evaporates and they begin to brown, about 5-7 minutes. Remove from the heat and let cool completely.
Coat a 9x13" baking dish with cooking spray. Arrange the bread cubes over the bottom. In a large bowl, beat the whole eggs, egg whites, milk, and mustard together until incorporated. Add the mushroom, onion, garlic mixture, broccoli, both cheeses, the sun-dried tomatoes, thyme, salt, and pepper and stir to incorporate. Pour the mixture over the bread, making sure the liquid saturates the bread. Cover with plastic wrap; refrigerate overnight or at least 8 hours.
Preheat the oven 350 degrees. Remove the plastic wrap from the strata and bake until the top forms a light brown crust, 60-70 minutes.