Combine water, brown sugar, and 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, carrot juice, and 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. spiced rum.
Top each with 3/4 cup chilled ginger beer. Garnish, if desired.