Preheat oven to 375°F. Grease a 2 1/2-quart baking dish with cooking spray. In a large skillet, melt butter over medium-high heat. Add onion and mushrooms; cook, stirring, until soft and golden, about 6 minutes. Remove from heat. Add spinach, mix well and season with salt and pepper.
Spread about 1/3 of bread cubes on bottom of baking dish, then 1/2 of vegetable mixture and 1/3 of cheese. Repeat with 1/3 of bread, remaining vegetables and another 1/3 of cheese. Top with remaining bread and cheese.
In a large bowl, whisk together milk, eggs and mustard. Season with salt and pepper. Pour over strata. Press down on bread mixture with back of a spoon to moisten.
Bake until strata is set and center is puffed and golden, 45 to 50 minutes. Serve immediately.