Lightly coat a 9x13-inch or similarly sized casserole dish with baking spray. Scatter the bagel pieces into the bottom of the dish. Scatter the ham (or bacon or sausage) over the top of the bagels. Sprinkle with 1/2 cup cheese.
In a large bowl, combine the eggs, milk, mustard, salt, and pepper until smooth. Pour the egg mixture over the bagels, then lightly press down on the bagels so that they absorb some of the egg. Cut the cream cheese into small pieces, then dot the pieces on top. Sprinkle on the 2 tablespoons fresh chives and remaining 1/4 cup cheese. Cover the casserole tightly with plastic or foil and refrigerate for 2 hours or overnight.
When ready to bake, remove the casserole from the refrigerator and let stand at room temperature while the oven preheats. Preheat the oven to 350 degrees F. Tent the casserole with foil and bake, covered, for 40 minutes. Remove from the oven, uncover, then return to the oven and continue baking until golden at the edges and set in the center, about 15-20 additional minutes. Let stand at room temperature for 10 minutes. Serve warm.
If your bagels are fresh or very soft, toast them in the oven first to dry them out and help them absorb the custard: Place two racks in the upper and lower thirds of your oven and preheat the oven to 300 degrees F. Spread the bagel pieces out in a single layer on two large baking sheets. Bake for 15 minutes, until browned and crisp. Use in the recipe as directed.
Casserole can be assembled and stored in the refrigerator up to 1 day in advance.
To freeze: Fully bake the casserole, then let cool and cover tightly. When ready to bake, let thaw overnight in the refrigerator, then reheat, covered, in a 350 degree F oven.