In a small saucepan, combine the fresh ginger slices, sugar, and water. Bring to a boil over medium heat then reduce heat to maintain a simmer. Continue simmering until reduced by half, about 15 minutes.
Pour syrup through a fine mesh sieve to remove ginger pieces. Let cool to room temperature then store in an airtight container in the refrigerator for up to 3-4 weeks.
Combine gin, lime juice, and ginger syrup in a cocktail shaker. Fill with ice and shake vigorously until completely chilled.
Strain into two copper mugs filled with ice. Top with ginger beer and garnish with mint and a lime wedge.
