Mince is not made from "hamburger". Hamburgers are made from mince. Mince, although used as a noun in this context, is a verb - to cut into small pieces. In Scotland it's also a slang adjective - not very good - "here, this mince is mince - could you no' mince it smaller?"
You mince meat to make mincemeat of it, but the mincemeat in mince pies doesn't actually have meat in it.
Graham Norton walked into a butcher's shop and said "can I have a mince round?" The butcher said "aye, but there's not much room."
I honestly can't understand why Americans have such difficulty with this stuff.
(Meanwhile, my dinner is cooking. Chicken breast stuffed with pancetta, spinach and brie, served with Jersey Royal potatoes and a rocket salad.)