Preheat the oven to 325°F (165°C). Line two baking sheets with parchment paper.
In a large bowl, beat butter and powdered sugar with a hand mixer on medium speed for 1 minute until creamy and pale. You can also pulse everything in a food processor.
Add vanilla and cardamom. Mix briefly.
Sift cocoa powder, flour, and salt over the bowl. Mix on low until the dough comes together in thick, sandy clumps. It should hold when squeezed. If dry, add 1 tsp cold water and mix once more.
Turn the dough out between two sheets of parchment. Roll to a quarter inch thick. If sticky, chill the slab 15 minutes.
Stamp 2-inch rounds with a cookie cutter. Reroll scraps once and cut again. Place rounds 1 inch apart on the prepared sheets.
Bake 14 to 18 minutes, swapping trays halfway, until edges turn matte golden and centers feel just set. Cool on the trays 3 minutes, then transfer to a wire rack until completely cool.