Slice 1 thin round of grapefruit; quarter the slice and set aside for garnish. Juice the remaining grapefruit halves into a pitcher (you should have about 1 cup juice). Wrap the ginger in a piece of cheesecloth or a kitchen towel and squeeze the juice into the pitcher. Stir in the beer, soda and bourbon.
Fill 4 glasses with ice. Pour in the cocktail and garnish with the reserved grapefruit slices.