Preheat oven to 325°F / 165°C (lower rack). Line and grease an 8-inch (20 cm) square pan with parchment paper.
Brown the butter – Melt the butter in a saucepan over medium heat. Whisk constantly as it foams and darkens. After about 10 minutes, it should smell nutty and have reddish bits at the bottom.
Mix the base – Remove from heat. Stir in the sugar, then cocoa powder, salt, baking powder, and espresso powder. Let the mixture cool for 5 minutes.
Add eggs one at a time, beating vigorously after each until fully incorporated.
Adjust thickness – Stir in just enough water (2 to 3 tablespoons) so the mixture becomes smoother but still thick. Stir in the vanilla.
Add the flour and beat firmly by hand for about 50 strokes.
By now, the mixture should be slightly warm but not hot. Fold in the chopped chocolate or chips.
Spread and swirl – Pour the batter into the prepared pan. Swirl the top gently with a knife or rubber spatula.
Bake for about 25 minutes, or until a toothpick inserted near the center comes out with just a few moist crumbs.
Cool completely in the pan before lifting out and slicing into 9, 12 or 16 squares.