Combine the cardamom, cinnamon and 3 cups water in a medium saucepan and bring to a bare simmer. Remove from the heat, stir in the sugar, and add the tea bags and 4 sprigs of the mint. Let steep for 6 minutes, then remove the tea bags. Let the tea cool to room temperature. Transfer to the refrigerator to chill. Strain the tea into a measuring cup, discarding the solids. Stir in the lemon juice.
Tear the remaining 8 sprigs mint and divide them evenly among 4 rocks glasses. Gently muddle the mint with the back of a spoon. Fill the glasses with ice, then divide the chilled tea evenly among the glasses. Add a shot (1 1/2 ounces) of rum to each glass and top off with a splash of sparkling water. Stir and serve.