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 small bowl, mix the instant coffee powder with hot water. Set it aside to cool.
In a large bowl, add the eggs and sugar. Using a hand mixer, whisk at high speed for at least 5 minutes, or until the mixture becomes thick, airy, and forms a ribbon-like consistency. If using a whisk, this step may take 10-15 minutes. This step is crucial for achieving the right texture.
Add the cooled instant coffee and vanilla extract to the egg mixture. Mix until well combined.
Sift in all-purpose 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.
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 your oven to 175°C (347°F).
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.
After 1 hour, do not mix the batter. Transfer the batter to a piping bag. Cut off the tip and pipe the batter into the molds, filling each about 80% full to allow space for rising.
Bake for 12-14 minutes, or until the edges are golden brown and the centers spring back when lightly touched.
Invert the pan onto the counter and transfer the warm madeleines to a wire rack to cool.