Place the diced pears, crushed cardamom pods, pear juice, honey, and cinnamon stick in a saucepan. Bring to a gentle simmer over medium heat and cook for 10 minutes to release the flavors.
Remove the pan from the heat and strain the mixture through a fine mesh strainer into a pitcher. Discard the solids and let the liquid cool to room temperature.
Stir the gin and lemon juice into the cooled pear base until fully combined.
Just before serving, add the sparkling water and stir gently. Fill glasses with ice cubes and pour the punch over the ice.
Garnish each glass with a fresh pear slice and a cinnamon stick.