To make the thyme simple syrup, combine the sugar, water, and thyme in a small saucepan. Bring to a boil over medium-high heat, stirring until the sugar dissolves. Remove from heat, and let the thyme simple syrup infuse until it’s completely cool. Remove and discard the thyme sprigs. Store in a jar in the fridge.

In a cocktail shaker, combine the gin, simple syrup, and lemon juice. Add ice, and shake for about 30 seconds, or until the outside of the cocktail shaker is frosty. Strain into a champagne glass. Top with champagne, tonic, or soda water. Slap a sprig of thyme against the palm of your hand to release the oils. Garnish the cocktails with thyme and lemon wedges.