Arrange bread in a 13 x 9-inch baking dish coated with cooking spray.
Combine 1/2 cup goat cheese and next 5 ingredients (through egg whites) in a medium bowl, stirring well with a whisk. Pour milk mixture over bread; top with remaining 1/2 cup goat cheese. Cover with foil coated with cooking spray. Refrigerate for 8 hours or overnight.
Preheat oven to 350°.
Remove bread pudding from refrigerator; let stand at room temperature 30 minutes. Uncover and bake at 350° for 40 minutes or until set and golden brown. Sprinkle with onions.