In a small saucepan, combine sugar and water then bring to the boil over medium heat. Stir until the sugar is dissolved, then stop stirring and increase heat.
Meanwhile, beat the eggs with an electric mixer until the syrup is done. Cook the syrup until the temperature reaches 115 degrees C according to a candy thermometer. Remove from heat and pour 1/4 cup of the syrup over the eggs, beating at high speed. Turn off the mixer and pour over another 1/4 cup of syrup, then turn it on and beat 10 seconds. Repeat until all the syrup is used, then beat until cool, about 5 minutes.
Beat in the butter 1 tablespoon at a time. Add the vanilla and beat until smooth and shiny, about 2 minutes. Add the espresso powder and beat until combined.