called one a “hamburger steak” and the other a “hamburger sandwich”.

White Castle continued calling it a “hamburger sandwich” for a long time until it became so ubiquitous, and hamburger steaks became so much less popular, that disambiguating was no longer needed.

So now we all say “a Hamburger” or “a burger” as short and shorter-hand for a “hamburger sandwich”, but we still use the article “a” to let you know we mean the sandwich item, as opposed to just ground beef (ie. “Go buy hamburger for dinner” would mean go get ground beef, not a hamburger (sandwich).

Sandwich according to Webster: “two or more slices of bread *or a split roll* having a filling in between” (emphasis added)

It certainly fits that primary definition.

For an additional test on your own:
Pull out the beef patty and put in pulled pork (or any other meat). What would you call that 100% of the time? Because a hamburger is just one very popular kind of sandwich.
Otherwise we’d have to say that a pulled pork sandwich, a Spicy chicken sandwich, etc aren’t sandwiches either.

It’s the ground beef (the “hamburger”) that makes it “a hamburger”. Not the bun. But even if you served it on toasted bread, it’s still a hamburger AND a still a sandwich.

But if you take it out of the bread/bun… it’s still “hamburger” but not “a hamburger”… because “*a* hamburger” is always a sandwich.

Thank you for coming to my Ted Talk.