For the Thousand Island Béchamel 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 béchamel then set aside and let cool.