Start by making the sugar syrup. Put the caster sugar in a small pan over a medium heat and pour in 50ml water. Stir, and bring to the boil.
Turn off the heat and allow the mixture to cool. Put 2 martini glasses in the fridge to chill.
Once the sugar syrup is cold, pour 1 tbsp into a cocktail shaker along with a handful of ice, the Absolut vodka, espresso and Kahlúa. Shake until the outside of the cocktail shaker feels icy cold.
Strain into the chilled glasses. Garnish each one with coffee beans if you like.
Put the sugar and 100ml water in a pan and heat gently, stirring, until the sugar dissolves. Cool. This will make more than you need but will keep in the fridge for a couple of weeks.
Put a handful of ice into a cocktail shaker followed by the vodka, coffee liqueur, hot espresso and 25ml of the sugar syrup. Shake vigorously for at least 15-30 seconds. Strain into a chilled martini or coupe glass and garnish with the coffee beans before serving.