Preheat oven to 400 degrees. In a 9-inch deep-dish pie plate, toss mushrooms, potatoes, and oil; season with salt and pepper. Roast for 20 minutes, tossing
Meanwhile, in a medium bowl, whisk together eggs, milk, mustard, hot-pepper sauce, 1/2 teaspoon salt, and 1/8 teaspoon pepper. Stir in cheddar and broccoli.
Pour egg mixture over mushrooms and potatoes, and stir to distribute them. Bake until puffed and set in the middle, and a paring knife inserted into the center comes out clean, 45 to 50 minutes.