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For the Thousand Island bechamel sauce, melt the butter in a medium saucepan over medium-high heat. Add the flour and cook, stirring occasionally with a wooden spoon, for about 2 minutes; do not allow to brown.
While whisking constantly, add the milk, salt, pepper and nutmeg. Cook the mixture, stirring occasionally, until it comes to a boil. Reduce the heat to medium-low and continue to cook until the sauce has thickened and any floury taste is gone, about 8 minutes.
Remove the sauce from the heat and add the ketchup and relish; transfer to a small heat-resistant bowl. Place a piece of plastic wrap on the surface of the bechamel, then set aside and let cool.
For the casserole, in a large mixing bowl, whisk together the eggs, cream, milk, salt, mustard, cayenne and nutmeg. Set the egg-cream mixture aside while you prepare the patty melt casserole.
Grease the bottom and sides of a 9-by-13-inch glass baking dish with 1 tablespoon butter.
In a mixing bowl, combine the cooled bechamel with the corned beef, sauerkraut and shredded cheese and set aside.
Using a serrated knife, cut the crusts off the bread. Place 8 slices of bread in the bottom of the baking dish. Spread half of the bechamel, then half of the sauerkraut and corned beef mixture over the bread. Top with the halved pre-cooked patties. Top with the remaining sauerkraut and corned beef, bechamel and 8 slices of bread.
Pour the egg-cream mixture evenly over the top. Top with a piece of parchment or wax paper and place a second baking dish on top to weigh down the casserole. Refrigerate for at least 1 hour and up to 2 hours. The bread should absorb most of the egg mixture.
Preheat oven to 350°F.
Slice remaining 4 tablespoons butter into 8 pieces and place one piece on top of each patty melt stack. Bake the casserole, uncovered, until puffed and golden, about 70 minutes. Let cool briefly before serving.