How to Make It
Combine 1 cup water, 1/2 cup packed dark brown sugar, and 6 large mint sprigs in a small saucepan. Bring to a boil over medium-high, stirring just until sugar is dissolved. Remove from heat, and cool completely (about 30 minutes). Discard mint sprigs.
Combine mint syrup, 1/2 cup cold-pressed carrot juice, and 1/4 cup fresh lime juice in a pitcher, and chill until very cold, about 2 hours.
For each serving, pour 6 Tbsp. of the mixture over ice in a 12-oz. glass, and stir in 3 Tbsp. spicedrum. Top each with 3/4 cup chilled ginger beer. Garnish with mint sprigs and shaved carrots.