Here’s why that’s wrong:
It’s because “burger” is a shortened form of the word “hamburger”, which is itself short for “hamburger sandwich”.
Hamburg is a place, and just like Frankfurt, or, well anywhere – say, Montreal – something from there is given the suffix, “-er”; Hamburger, Frankfurter, or Montrealer. (See also: New Yorker, East Coaster, West Coaster, etc, etc, ad nauseam.)
“burger” as a word therefore doesn’t really make very much linguistic sense; it’s a slang term, not an actual culinary term. The fact that some people call ground beef “burger” (or the patty itself) doesn’t legitimize the usage, as it once again is merely a slang term.
Therefore it is inaccurate to suggest that there is a single ‘proper’ usage for the term “burger”.
Even the dictionary does not agree with you:
[https://www.dictionary.com/browse/burger](https://www.dictionary.com/browse/burger)
>noun
>
>\-a hamburger.
>
>\-a food patty, or patty on a bun, containing ingredients other than beef:*veggie or turkey burgers*.
Now at first, one might be inclined to hyper-focus on the word ‘patty’. However, the dictionary still doesn’t agree with you, because:
[https://www.dictionary.com/browse/patty](https://www.dictionary.com/browse/patty)
>noun, plural pat·ties.
>
>\-any item of food covered with dough, batter, etc., and fried or baked:*oyster patties.*
>
>\-a thin, round piece of ground or minced food, as of meat or the like:*a hamburger patty.*
>
>\-a thin, round piece, as of candy:*peppermint patties.*
>
>a little pie; pasty.
As one can clearly see, a breaded and fried chicken cutlette meets the first criteria of the word, ahead even of “hamburger patty”; so please don’t suggest that the “food patty” definition of burger is a “secondary definition”.
As for your claims about your version being true in North America; I can’t speak for the United States, as I have only visited a few times – but I have lived in 5 different Provinces in about 11 different cities across Canada, from 1962 until present day. “Chicken burger” is common coast to coast, and while some US-based restaurant chains call them “chicken sandwich” and no one bats an eye, it is similar to the “zee vs. zed” discussion we have, where both are acceptable but there is only one that is actually correct, and north of the 49th parallel it’s “chicken burger”.
(It’s “zed” if you’re Canadian, by the way.)
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[*burger sauce*](https://old.reddit.com/r/iamveryculinary/comments/omkfsb/one_comment_in_food_annoys_a_mod_and_seemingly/h5lwvbo/)