Make the Shortbread Dough:
If using fresh cardamom pods, grind them into a fine powder.
In a stand mixer or using a hand mixer, beat the softened butter and powdered sugar on low speed until combined. Increase the speed to medium and continue to beat until the mixture is light and creamy, about 30 seconds. Scrape down the sides as needed.
Add the ground cardamom and flour, then mix on low speed until the dough comes together into medium-sized chunks and there are no visible flour streaks. Do not overmix.
Using a spatula or your hands, press the dough together, form it into a disk, and wrap it tightly in plastic wrap. Refrigerate the dough for 30 minutes.
Prepare for Baking:
Preheat the oven to 325°F (160°C) and line a baking sheet with parchment paper.
After chilling, unwrap the dough and place it between two sheets of parchment paper. Roll it out to a thickness of ½ inch. Cut the dough into shapes using cookie cutters.
Place the cut cookies onto a parchment-lined plate and freeze for 10-15 minutes.
Bake the Shortbread:
Transfer the chilled cookies onto the prepared baking sheet. Bake for 20-22 minutes, or until the edges are lightly golden.
Remove the cookies from the oven and let them cool on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.
Make the Orange Glaze:
In a small bowl, whisk together the powdered sugar, orange juice, and milk until smooth.
Once the cookies have cooled, dip the top side of each cookie into the glaze, then turn them right-side up.
If desired, sprinkle with orange zest for extra flavor and decoration. Let the glaze set before serving.