In a medium saucepan, combine the sugar, water, lime peel, and ginger. Bring to a rolling simmer over medium-high heat until the sugar is completely dissolved. Then reduce to medium-low and continue simmering, stirring occasionally, for 20 minutes.
Remove from heat, add in the sprig of mint and lime juice, and set aside to cool for at least 1 hour.
Once cool, strain through a fine-mesh stainer, and pour into an airtight container. Store in the fridge for up to 2 weeks!
In a large glass or copper mule mug, add lime juice. Fill the glass with ice.
Top with vodka, ginger syrup, and ginger beer. Gently stir, then top with a lime wedge or spiral and a sprig of mint. Serve at once!
