Preheat the oven to 325°F. Lightly butter an 8x11 oval baking dish.

Melt the butter and set it aside to cool slightly.

In the bowl of a stand mixer fitted with the paddle attachment, beat the eggs and sugar on medium-high speed for 5–10 minutes, until very thick and pale yellow.

Meanwhile, sift the cocoa powder and flour together and set aside.

Reduce the mixer speed to low and add the vanilla extract, and the cocoa-flour mixture. Mix just until combined.

With the mixer still on low, slowly pour in the cooled butter and mix just until incorporated.

Pour the batter into the prepared baking dish. Place the dish inside a larger baking pan and carefully add very hot tap water to the outer pan so it comes halfway up the sides of the baking dish.

Bake for exactly 1 hour. A cake tester inserted about 2 inches from the edge should come out mostly clean. The center will look very underbaked—this is intentional and gives it a pudding-like texture. Let cool slightly, then serve warm with vanilla ice cream.