Just saw Guy Fieri make this on Guy's Big Bite, Episode: Kentucky Klassics and wanted to make it not using alcohol. So here it is. The recipe makes 2 cups of simple syrup, so you can easily make more than 1 drink.
Trim the lemongrass stalks so you have the bases only. Bash with the back of a knife (or a meat mallet) so the oils and aroma release more easily.
Combine the lemongrass, sugar, 2 cups cold water and ginger together in a medium saucepan and set over medium heat. Simmer until the sugar dissolves. Reduce the heat and allow to sit so the lemongrass and ginger steeps in the syrup. Once the syrup is fragrant, 30 to 40 minutes, strain and use. Yield: 2 cups.
Add the mint and ice to a cocktail shaker and muddle. Add the Lemongrass-Ginger Simple Syrup and shake vigorously to combine. Serve on the rocks in a Collins glass. Top with club soda, a squeeze of lime juice and garnish with a sprig of mint.