In a medium bowl, whisk together the flour, ground cardamom, and salt.
Using a stand mixer with paddle attachment or electric hand mixer, cream butter until light and creamy, about 2 minutes. Scrape down sides of bowl using a rubber spatula.
Add confectioner's sugar and vanilla, and mix until well combined.
Add flour mixture and mix for another 1 to 2 minutes, until dough comes together.
Transfer dough onto large piece of plastic wrap. Wrap dough and use a rolling pin to flatten into a disk. Refrigerate for 30-60 minutes, until the dough is firm enough to roll out.
Preheat the oven to 350 degrees F. Line 3 baking sheets with parchment paper.
Place chilled dough onto a lightly floured work surface and roll the dough to a little less than ½-inch thickness. Use cookie cutters to cut desired shapes (if you have a cookie stamp, use it here).
Place the cookies 1 inch apart on the baking sheets. Refrigerate for 30 minutes or place in freezer for 10 minutes.
Bake for 8-10 minutes, until lightly browned around the edges and center is just set.
Let cookies cool on baking sheets 2-5 minutes; transfer to a wire rack, and let cool completely.