Let me begin by saying that I have NO IDEA what makes this cake “French.” All I know is that, when I got the recipe years and years ago, that’s what it was called, so I’m running with it.
In the pan of a double-boiler, start melting 9 ounces of good quality chocolate with 2 sticks of butter (I know, I know, but trust me…). Once you’ve got it mostly melted, stir in a scant half cup of white sugar and stir until it’s all smooth.
While that’s happening, beat together 5 eggs with a tablespoon of flour (yes, that’s all; just one tablespoon). When the chocolate, butter, sugar mess is all nice and smooth, stir in a tablespoon or so of vanilla, then mix it all together with the eggs. Scrape it all into the prepared springform, pour hot water into the baking dish, and park it in the center of your oven for about 20-25 minutes, or until set but still soft. Let it cool completely on a wire rack (I usually pull the springform ring off it if, just in case any water got through my tin foil defenses).
Whether you choose to fridge it or not before serving it in little slivers (be careful; it’s rich) is entirely up to you. This needs nothing in the way of extra, but I usually sieve a little powdered sugar over the top before I serve it.