Grind cardamom pods in a spice grinder or food processor until broken into chunks.

Combine water and cardamom in a small pot and bring to a boil.

Remove from heat and let steep for 10 minutes.

Add sugar and stir until dissolved.

Let steep at room temperature for 2 hours, then fine strain.

Combine syrup, lemon juice, honeydew juice, and Pimm's in a cocktail shaker.

Fill with ice.

Shake until well chilled, about 15 seconds.

Strain into an ice-filled Collins glass.

Top with ginger ale.

Garnish with lemon slice and melon ball if desired.