Gently melt the butter using either a double-boiler or microwave, allowing it to reach a temperature of 45-60°C. Allow it to rest.
In a mixing bowl, blend together eggs and icing sugar.
Sift in cake flour, baking powder, and cocoa powder, then mix until well combined.
Gradually incorporate the melted butter in three separate additions, ensuring thorough mixing before each addition.
Introduce Kahlúa to the batter and combine thoroughly.
Cover the batter and let it chill in the refrigerator for 30-60 minutes, ensuring not to exceed this timeframe to prevent the butter in the batter from solidifying.
Preheat the oven to 446°F (230°C).
Melt approximately 2 tablespoons of butter and delicately brush the pan using a pastry brush. Evenly sift cake flour onto the pan, ensuring to remove any excess flour.
Place the batter into a piping bag and pipe it into the pan until it reaches 80% full.
Bake for 2 minutes at 446°F (230°C), then briefly open the oven door for 5-7 seconds. Lower the temperature to 180°C and continue baking for an additional 9 minutes. The madeleines are ready when the tops spring back upon a gentle press with your finger.
Invert the pan onto the counter and transfer the warm madeleines to a wire rack to cool.
In a mixing bowl, whisk mascarpone cheese and sugar.
Add whipping cream in three parts, mixing well before each addition.
Add Kahlúa and mix until fully incorporated.
Use a round piping tip to create a small hole in the center of the madeleines.
Transfer the mixture into a piping bag and fill the hole and spread onto the madeleines. Dust with cocoa powder, and your delightful treats are ready to be served!