For the demerara sugar syrup: Combine the demerara 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.
For the lavender lemon drop: Add the gin, limoncello, lemon juice and 1/2 ounce of the demerara sugar syrup to a cocktail shaker, add large ice and shake vigorously. Add the Moscato d'Asti, tumble-roll once to mix throughout and double-strain into a martini glass with a small sprig of lavender. Garnish with a lemon wedge and serve (do not muddle the lavender).
Keep the remaining demerara sugar syrup covered in the refrigerator up to 1 month.
Remove the peels from the lemons carefully, making sure not to include any of the pith. Place the peels in a large container and add the vodka (reserve the vodka bottle for later). Infuse until the alcohol strips the oils and color out of the peels, 12 to 24 hours.
Meanwhile, make the simple 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.
Fill the vodka bottle with about 27 ounces of the infused vodka and 7 ounces of the simple syrup and place in the freezer. Serve ice cold in chilled small copita or shot glasses.