Rinse a chilled cocktail coupe with absinthe, then garnish the glass with a tarragon sprig.
Add the gin, Demerara Sugar Syrup, lemon juice, Green Chartreuse and remaining tarragon sprig to a cocktail shaker. Muddle the tarragon, add large ice cubes and shake vigorously. Double-strain into the prepared glass and serve.
Demerara Sugar Syrup:
Combine the sugar and 1 cup water in a small saucepan and bring to a boil. Cook, stirring, until the sugar dissolves. Remove from the heat and let cool completely. Keep covered in the refrigerator up to 1 month.