Melt the butter in a saucepan until it colors and starts to smell slightly nutty. This is how to make the beurre noisette. Let the beurre noisette cool.
Beat the eggs with sugar in a large bowl for 5 minutes until light and fluffy. Add the beurre noisette to this little by little.
Then add the espresso powder, vanilla extract, flour and a pinch of salt and mix in. Put the batter in the fridge for at least 1 to 2 hours.
Meanwhile, preheat the oven to 200 degrees Celsius / 390 Fahrenheit top and bottom heat.
Grease the tin / mold and dust very lightly with a little flour.
Divide the batter so that they are just not completely full (they will expand slightly during baking).
Bake the madeleines for about 8-10 minutes until lightly golden brown and cooked.
Let the madeleines cool and, in the meantime, make the icing by mixing the espresso with the icing sugar little by little. You don't have to use all the espresso, just add a teaspoon each time until you get a nice thick paste.
Dip the madeleines in this paste and let the icing harden. Then enjoy your madeleines!