You're all wrong. The historical stages of precedence go: Frankfurt, Hamburg, Lord Sandwich. The hamburger is a member of the class frankfurter, and the sandwich is a further step down the taxonomic tree. All three are processed protein surrounded ("sandwiched," or more properly "franked" like postage stamps) by bread, with various condiments.
That's assuming, of course, that we want to factor in time, as in the modern revisions of the Linnaean system. We could do a straight Linnaean taxonomy, in which case we'd have to decide which system of these foods is the one studied (bread, meat, processing, cooking, condiments...), or a polythetic taxonomy studying all the many factors present in a wide distribution of all these foods, leading to definitions based on statistical variation. Or of course there's always Goethe's morphological system....