Preheat the oven to 350 degrees F. Butter an 8'' or 9'' springform pan (or cake pan with a removable bottom). Lining the bottom with buttered parchment helps to remove the cake as well.
Pulse the dark chocolate in a food processor until it's very fine. Add the sugar and pulse until combined.
Prepare 3/4 cup of strong coffee and add it piping hot into the food processor while it's running. Blend until all the chocolate is completely melted and the mixture is smooth.
Finally, add in the beaten eggs and vanilla, then process until smooth. (Be sure the eggs are already beaten before adding, or you might end up with a trail of egg white in the final cake.) The batter will be quite thin.
Pour the batter into a buttered springform pan and bake at 350 for about 60 to 75 minutes. Keep in mind that smaller pans mean a thicker cake, and longer bake time. Test for doneness with a toothpick in the center.
When the cake is done, remove it from the oven and allow it to cool at room temperature. It will deflate, and that's normal. The finished cake is quite thin.
Refrigerate for at least 3 hours before serving.
Notes
For the market, we would make a thick chocolate ganache with melted dark chocolate and heavy cream and pour it over the cake before refrigerating. That's optional but extra decadent. This is a very rich cake, so cut into tiny slices. One cake makes about 12 slices.