Melt the unsalted butter using the double-boiler method, or by microwave. Set it aside to cool.
In a small bowl, mix instant coffee powder and hot water. Set it aside to cool.
In a bowl, add egg and sugar. Use a hand mixer to whisk at high speed for at least 5 minutes, or until the mixture becomes thick, airy, and can form ribbon. If you are using a whisk, this may take 10-15 minutes. (This step is important!)
Add instant coffee and vanilla extract to the mixture. Mix until combine.
In another bowl, combine sifted all-purpose flour, salt, and baking powder.
Pour in half of the dry ingredients into the wet mixture. Fold in lightly and mix until combined. Pour the remaining dry ingredients. Fold in lightly and mix until combined.
Add approx 1/4 cup of the mixture to the melted butter. Mix until it fully incorporates. This may take a few minutes.
Pour the butter mixture with the remaining batter. Mix until combine.
Cover the bowl with plastic wrap. Refrigerate for 1 hour.
Preheat the oven at 175 °C or 347 °F.
Prepare some melted unsalted butter. Brush the butter to the madeleine mold.
After 1 hour, do not mix the batter! Transfer the batter directly to a piping bag.
Cut off the tip and pipe batter into a baking pan up to three quarters, so the madeleines have enough space to rise.
Bake for 12-14 minutes.