Spray a 9x13-inch baking dish with cooking spray. Spray one sheet of aluminum foil with cooking spray.
Spread bread cubes in the bottom of the prepared baking dish. Layer bread cubes with sauerkraut and beef strips, then pour dressing over top. Cover with the aluminum foil, sprayed-side down.
Bake in the preheated oven for 20 minutes. Remove and discard foil; sprinkle Swiss cheese over casserole. Continue baking until cheese is melted and bubbly, about 10 more minutes.