Preheat the oven to 325℉, grease a 2-quart baking dish with butter, and place it in a deep roasting pan, or similar. Pro Tip: Use the butter wrappers to grease the dish.
Add enough hot tap water to the roasting pan so that it's halfway up the sides of the baking pan the batter will go in.
Melt 1 cup unsalted butter in a medium pot, then remove from heat and allow to cool while you prepare the rest of the batter.
Add 4 x-large eggs and 2 cups granulated sugar to the bowl of a stand mixer fitted with a paddle attachment, or to a large bowl with a hand mixer, and beat on speed 6 for 5 to 7 minutes, scraping down the sides and bottom of the bowl with a rubber spatula every couple of minutes. The mixture should be thick and pale yellow.
Slice the 1 vanilla bean in half with a paring knife, then use the flat edge of the knife to scrape out the seeds (vanilla caviar). Mix in the vanilla seeds and ¾ teaspoon coffee extract until combined.
Sift ¾ cups unsweetened cocoa powder, ½ cup all-purpose flour, and ¾ teaspoon fine sea salt into the egg mixture and then mix just until combined. Scrape down the sides and bottom of the bowl.
With the mixer on low speed, slowly pour the melted butter into the bowl, ensuring you scrape any remaining butter out of the pan and into the bowl.
Scrape down the sides and bottom of the bowl with a rubber spatula and fold the batter for 15 to 30 seconds to combine thoroughly.
Pour the batter into the prepared baking dish and smooth out the top with the rubber spatula.
Place the baking dish with the batter in the water bath and bake for 45 to 60 minutes. A cake tester, when inserted into the pudding 2 inches from the edge, should be mostly clean when done; the center will be very jiggly and saucy.
Let the cake rest for 5 to 10 minutes before scooping out and serving in a bowl with ice cream. You can also serve with fresh berries or a drizzle of heavy cream instead of ice cream.