Remove peel from pineapples; cut peel into 3-inch pieces (reserve pulp for another use). Combine peels and 6 cups water in Dutch oven; bring to a boil. Reduce heat; simmer 30 minutes. Drain in a colander, reserving 4 cups cooking liquid; press peels to release moisture. Discard peels.
Combine ginger, lemongrass, and 2 tablespoons water in a food processor; process until finely chopped. Place ginger mixture on a dampened double layer of cheesecloth. Gather edges of cheesecloth together; tie securely. Place cheesecloth bag and 4 cups water in a medium bowl; let stand 30 minutes. Strain reserved pineapple cooking liquid and ginger mixture into a large pitcher; discard cheesecloth bag. Add sugar; stir well. Chill. Serve over ice. Garnish with mint sprigs, if desired.