For the Ginger-Infused Bluecoat Gin:
Combine the peeled ginger with the gin in a blender and puree. Using a fine mesh strainer, separate the pulp from the gin. Store the gin in a tightly sealed bottle.
For the Cocktail:
Pour the beer into a Pilsner glass. In a shaker, combine ginger-infused gin, lime juice, and simple syrup. Add ice, shake, and strain into the glass. Garnish with the twist of lemon.