If you are talking about biftek hache or just plain patties of ground meat, chances are they’re made the “French” way with suet wrapping, more fat than you’ll find in a burger, sometimes eggs or finely diced onions and garlic, and cooked in broth before being finished on the grill.
I don’t know either of the two restaurant’s way of making them, but I am familiar with the question. Unlike the American purism (ground beef, period, seasoning on the outside if at all), French ground beef patties are usually seasoned throughout and oftentimes made with the ingredients listed above. Rubbed sage or tarragon are another possibility.
So in conclusion, while I don’t know the two restaurants you refer to, “French” hamburgers are made with more fat, usually cooked in broth before being grilled, contain thyme, tarragon, or sage as seasonings, and often include eggs and onions for taste and structure.