Melt the butter in a saucepan and add the flour, stirring constantly. Add the milk little by little and stir well, so that all the lumps disappear. When all the milk has been added, let it boil for a little while and then remove from the heat and add the cheese and stir completely into the sauce.
Grill the beef burgers on both sides over the hot Binchotan until they are just cooked inside and have a nice crust on the outside. Cut the brioche buns in half and heat briefly over the Binchotan.
Place the burger between the buns and finish with a few slices of gherkin, the cheddar cream and a little ketchup and mustard.