Hamburgers are just a strange thing to be fixated on if someone considers a fantasy game as "Medieval".
Like why would burgers be the problem? Why not zippers? or the fact the color blue is widely available? Or large sailing ships? Pasta with tomato sauce? Widely available medicine?
In the literal dark ages where people simply couldn't afford meat and were struggling to just get by. Of course they wouldn't have the luxury of ground beef. The only thing stopping it was people not having the time/resources to make them.
Even further back when people DID have free time, Romans had a burger (
Link), so it's just a weird hangup. They even had restaurants with take out.