Preheat the oven to 325°F and generously butter a 9x12 inch baking dish.
Melt the butter and set it aside to cool slightly while you prepare the batter.
In the bowl of a stand mixer with the paddle attachment, beat the eggs and sugar on medium-high speed until the mixture becomes very thick, pale, and glossy. This will take between 5-10 minutes.
In a separate bowl, sift together the cocoa powder and flour.
Reduce the mixer speed to low and add the vanilla bean paste to the sugar mixture, followed by the cocoa and flour mixture. Mix just until combined.
With the mixer still running on low, slowly pour in the cooled melted butter and mix until fully incorporated.
Pour the batter into the prepared baking dish and place it inside a larger baking pan.
Carefully pour hot tap water into the outer pan until it reaches about halfway up the sides of the baking dish.
Bake for 1 hour. A tester inserted near the edge should come out mostly clean, while the center will still look very soft and under-baked.
Remove from the oven, allow to cool for 5-7 minutes, and serve warm with vanilla ice cream.