Preheat the oven to 350 degrees F. Butter a 13-by-9-inch baking dish or a 2-quart casserole with butter or coat it with cooking spray.
In a skillet over medium heat, cook the onions, stirring occasionally, until golden, about 3 minutes, then add the mushrooms. Continue to cook until softened, then add the green beans and garlic and cook until fragrant, 2 to 3 minutes.
In a large bowl, toss the mushroom mixture with the stale bread until well mixed, then transfer to the baking dish and spread into an even layer.
Whisk the eggs in the bowl until pale yellow, then whisk in the cream, milk, nutmeg, 1/2 teaspoon salt and 1/4 teaspoon pepper. Slowly pour over the bread mixture 1 cup at a time, giving it time to absorb between additions. Cover with aluminum foil and bake until it is puffed and set, about 45 minutes.
Crumble the fried onions and green bean chips together. Remove the foil from the pudding and spread the crumbs across the top. Bake until the onions are golden brown, about 5 minutes more.