Combine blueberries, water, cinnamon stick, honey, ginger and cardamom pods (if using) in a large saucepan. Bring to a boil, stirring occasionally. Reduce heat and simmer, stirring, until most of the blueberries have burst, 1 to 2 minutes. Remove the cardamom pods and cinnamon stick. Puree the soup in 2 batches in a blender until smooth (use caution when pureeing hot liquids). Place a fine sieve over the pan and pour the soup through it back into the pan, straining out any solids. (Discard the solids.)
Whisk cornstarch and milk in a measuring cup until smooth. Whisk into the blueberry mixture. Bring the soup to a boil over medium heat, stirring. Boil, stirring constantly, until the soup thickens slightly, about 1 minute. Remove from the heat and let cool for 10 minutes. Transfer to a bowl, loosely cover and chill until cold, at least 5 hours or up to 2 days.
Just before serving, whisk 1 cup sour cream into the soup and ladle into bowls; top each serving with 1/2 teaspoon sour cream and swirl decoratively into the soup. Garnish with additional blueberries, if desired.