In a small saucepan over medium heat, combine the granulated sugar and filtered water.
Stir occasionally until the sugar fully dissolves. This usually takes about 3-5 minutes.
Once dissolved, remove the saucepan from heat and let it cool. Your simple syrup is ready!
In a blender, add the fresh blackberries and caster sugar.
Blend until smooth. If you prefer a less pulpy texture, you can strain the mixture through a fine-mesh sieve.
Mix in the crème de cassis and lime juice. Your berry puree is now all set for the cocktail!
Take the superfine sugar and spread it on a small plate.
Rub the rim of your martini glass with a lemon or lime wedge.
Dip the glass rim into the sugar to create a sugared edge.
In a shaker filled with ice, combine the gin (or vodka), triple sec, freshly squeezed lemon juice, and about 2 tablespoons of the berry puree.
Shake well until chilled; about 15-20 seconds should do.
Strain the mixture into the prepared martini glass.
Optional: Add a splash of soda water for a lighter finish.
Garnish with a lemon wedge, a sprig of fresh mint, and enjoy immediately!