Melt the butter in the microwave and set aside to slightly cool.
Using a handheld or stand mixer fitted with a whisk attachment, beat the eggs and sugar together on high speed for 8-10 minutes or until the mixture is thick, very pale, and forms ribbons when you lift the beaters. Add in the vanilla extract and beat until combined.
In a small bowl, whisk together the flour, hazelnut flour, baking powder, and salt.
Using a spatula, gently fold the flour mixture into the egg mixture. Then gently fold the melted butter into the batter. It will take a minute or two to fully incorporate, so just keep gently mixing it together until fully combined. The batter should be thick and shinny.
Cover the bowl and let the batter chill in the fridge for 45 minutes.
During the last few minutes of chilling, preheat the oven to 350°F.
Lightly brush your madeleine pan with 1-2 tablespoons of melted butter, then scoop a generous tablespoon of batter into the center of each well.
Bake for 10-12 minutes or until the madeleines are lightly browned.
Transfer to a wire rack and let cool.
Dip the cooled madeleines in melted dark chocolate and garnish with chopped hazelnuts. Eat with coffee and enjoy!
Madeleines are best eaten the same day but will keep in an airtight container at room temperature for 2-3 days.