To make the simple syrup:
Combine the water, sugar, chopped ginger and mint leaves in a small saucepan over low heat. Cook, stirring occasionally until the sugar is completely dissolved.
Let the syrup simmer for about 15 minutes over low heat then remove and let it cool for about 30 minutes. Strain the ginger and the mint from the syrup.
The simple syrup can be made ahead and refrigerated until you’re ready to mix up the cocktails.
To make the cocktails:
Pour the lemon juice, simple syrup, gin and ice into a blender. Process, starting on low for about 5 seconds to break up the ice then on high for an additional 15 seconds.
Fill two glasses with ice, sliding a slice of lemon along the side of the glass for garnish if desired.
Pour the gin mixture between the two glasses and top with the beer, splitting evenly between drinks, stirring to mix the spirits together.
Top with a sprig of fresh mint and serve!