Everything Bagel Breakfast Casserole

leftover everything bagels, halved horizontally and cut or torn into bite-sized pieces (see notes if your bagels are fresh or very soft)

cooked ham, bacon, or sausage, cut or crumbled into small pieces

freshly grated sharp cheddar cheese, divided

large eggs

milk

ground mustard

kosher salt

black pepper

chopped fresh chives, plus additional for sprinkling on top

reduced-fat cream cheese (or regular cream cheese; do not use fat free)