I am not 100% on this but hamburger comes from the melding of the name of the town in Germany Hamburg where the dish can trace it's modern root's although I know there was a roman dish like a hamburger but it was served with fish sauce with a chunk of bread the source i site for this is the roman cook book written by MARCUS GAVIUS APICIUS. And the main meat it was made from was I belive venison rather than beef.
Edit here we go found it:
ISICIA OMENTATA (a kind of Roman Burgers)
(Apic. 2, 1, 7)
Ingredients:
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500g minced meat
1 french roll, soaked in white wine
1/2 tsp freshly ground pepper
50ml Liquamen (can be replaced by 1/2 tsp salt + a little white wine)
some stone-pine kernels and green peppercorns
a little Caroenum
Baking foil
Instructions:
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Mix minced meat with the soaked french roll. Ground spices and mix into
the meat. Form small burgers and put pine kernels and peppercorns into
them. Put them into baking foil and grill them together with Caroenum.