Heat oven to 400°F. Oil 3-quart baking dish and piece of foil. Heat oil in large skillet on medium-high. Cook half of mushrooms, tossing occasionally, until golden brown, 4 to 6 minutes; transfer to plate. Repeat with remaining mushrooms, adding more oil if necessary. Meanwhile, in large bowl, whisk together eggs, milk, mustard, nutmeg, and 1 teaspoon each salt and pepper. Add bread and toss to coat. Fold in mushrooms, spinach, shallots and all but 1/2 cup cheese. Transfer mixture to prepared dish, sprinkle with remaining cheese, cover with foil and bake 30 min. Remove foil and broil until golden brown, 2 to 4 minutes.