Combine first 3 ingredients in a medium saucepan over medium-high heat; bring to a boil. Cover, reduce heat to low, and simmer 10 minutes or until peaches are tender. Remove from heat; cool. Discard cinnamon stick.
Place half of peach mixture in a blender or food processor; process until smooth. Pour pureed mixture into a large bowl; repeat procedure with the remaining peach mixture. Stir in 3 tablespoons honey, juice, and 1/4 teaspoon vanilla. Chill for 1 hour.
Ladle 1/2 cup soup into each of 6 small bowls; top each serving with about 2 1/2 tablespoons blackberries. Garnish with fresh mint leaves, if desired.