Melt the butter and chocolate together gently, then allow the mixture to cool until lukewarm.
Beat the eggs and sugar for 3–4 minutes, until pale, thick, and increased in volume.
Mix in the vanilla.
In a separate bowl, whisk together the flour, baking powder, and salt.
Fold the dry ingredients gently into the egg mixture.
Stir a few spoonfuls of batter into the chocolate-butter mixture, then fold this mixture back into the remaining batter. This helps incorporate the chocolate without deflating the eggs.
Cover and refrigerate for at least 1 hour or overnight.
Preheat the oven to 190°C/375°F conventional or 175°C/350°F fan-assisted.
Generously butter and lightly flour the madeleine molds, tapping out the excess.
Fill each mold about three-quarters full.
Bake standard madeleines for approximately 9–12 minutes, until the centers are set and spring back when lightly pressed.
Rest for 1–2 minutes, then unmold and transfer to a wire rack.