Whisk eggs in a large bowl. Add milk, hot sauce, 1/2 teaspoon salt and 1/8 teaspoon pepper; whisk until blended. Add bread and stir to coat well. Cover and refrigerate for 30 minutes.

Meanwhile, preheat oven to 350 degrees F. Coat an 8-by-8-inch (or similar 8-cup) ceramic or glass baking dish with cooking spray. Put a kettle of water on to boil for the water bath.

Cook spinach according to package directions. Drain, refresh with cold water and squeeze out excess moisture. Set aside.

Heat oil in a large nonstick skillet over medium heat. Add onion and bell pepper; cook, stirring often, until tender and golden, 4 to 5 minutes. Add mushrooms and garlic; stir to blend. Reduce heat to medium-low, cover and cook until tender, about 5 minutes. Uncover and increase heat to medium. Cook, stirring, until any excess moisture from the mushrooms has evaporated and mushrooms begin to brown, about 5 minutes. Add the spinach, remaining 1/4 teaspoon salt and 1/8 teaspoon pepper; cook, stirring, 2 minutes.

Add the mushroom mixture and 1/2 cup cheese to the egg mixture; stir until blended. Scrape into the prepared baking dish; sprinkle the remaining 1/2 cup cheese evenly over the top. Place the baking dish in a larger pan, put in the oven and pour boiling water into the larger pan to about halfway up the sides of the baking dish. Bake until the pudding is set in the center, 50 to 60 minutes. Let cool slightly; serve hot or warm.