Preheat oven to 350°F. Grease a 13"x9" baking dish with butter or nonstick cooking spray.
Spread half of the 1-inch cubed rye bread evenly across the bottom of the prepared baking dish. It should nearly cover the bottom of the pan.
Layer half of the corned beef or pastrami over the bread cubes.
Disperse the prepared sauerkraut and pickles evenly on top.
Sprinkle half of the shredded Swiss cheese over the pickles.
Top with the remaining 1-inch cubed rye bread followed by another layer of corned beef or pastrami. Press down so that the picture compacts slightly.
Sprinkle the remaining shredded Swiss cheese over the top.
In a small bowl, combine milk, Thousand Island dressing, eggs, butter, Dijon mustard, and caraway seeds. Pour the mixture evenly over the casserole.
In a bowl, toss the tiny rye bread cubes with the remaining butter. Sprinkle on top of the casserole.
Spritz a sheet of aluminum foil with non-stick spray or oil. Cover the baking dish tightly with the foil, greased side down. Bake 40 minutes. Remove the foil and bake for an additional 10-15 minutes, or until the cheese is melted and bubbly and bread crumbs are toasted.
Allow the casserole to rest at least 5 minutes prior to serving. Garnish with fresh parsley or additional chopped pickles, if desired.