For the 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.
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-inch x 13-inch glass baking dish with 1 tablespoon of butter.
In a mixing bowl, combine the cooled Thousand Island bechamel recipe with the corned beef, sauerkraut and shredded cheese; set aside.
Using a serrated knife, cut the crusts off the bread. Place eight slices of bread down in the bottom of the casserole dish. Spread half of the bechamel and half of the sauerkraut and corned beef mixture down over the bread. Top with the halved pre-cooked patties. Top with remaining sauerkraut and corned beef, bechamel mixture and eight slices of bread.
Pour the egg-cream mixture evenly over the top. Top the casserole with a piece of parchment or wax paper and place a second casserole 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.
Pre-heat dthe oven to 350°F. Slice the remaining 4 tablespoons butter into eight pieces and place one piece on top of each patty melt stack. Bake the casserole, uncovered, until puffed and golden, about 70 minutes. Let the casserole cool briefly before serving.