Preheat the oven to 325°F (163°C) and grease a 9×13-inch baking dish with unsalted butter.

In a large mixing bowl, beat the eggs and granulated sugar on medium speed for 5 to 10 minutes until pale and thick.

Slowly stir in the melted butter and vanilla extract into the egg and sugar mixture until smooth.

Sift together the cocoa powder, flour, espresso powder (if using), and salt in a separate bowl.

Gently fold the dry ingredients into the wet ingredients until just combined.

Pour the batter into the greased baking dish, spreading it evenly. Place this dish into a larger pan filled with hot water.

Bake for 55 to 65 minutes, until edges are set and a toothpick comes out with moist crumbs.

Let the pudding cool for about 10 minutes before serving; best served warm with vanilla ice cream or whipped cream.