Heat butter in a sauté pan over medium heat. Add leeks, mushrooms, 1/2 teaspoon salt, 1/2 teaspoon pepper, brown sugar and nutmeg. Cook, stirring occasionally, until leeks are soft and golden brown, about 30 minutes.
Place bread cubes on a baking sheet and bake 15 minutes or until dry and lightly browned.
Preheat oven to 350F. Grease a shallow 4-quart baking dish.
Combine leek mixture, bread cubes and remaining ingredients. Mix well and let stand 20 minutes. Press mixture into prepared pan and sprinkle with remaining salt and pepper. (You may cover and refrigerate the bread pudding up to 4 hours before baking.)
Bake about 1 hour, or until pudding is set and golden brown on top. Remove from the oven and let cool 10 minutes before serving.