In a mortar, using a pestle, crush the cardamom pods until they open. Free all of the black seeds from the pods and discard the pods. Crush the seeds until finely ground. You should have about 1 tablespoon crushed cardamom seeds..
In a food processor, combine the flour, salt, crushed cardamom seeds, brown sugar, and granulated sugar and pulse a few times to mix. Scatter the butter and cream cheese over the flour mixture and pulse until most of the butter and cream cheese is no longer visible and is roughly incorporated, about 30 seconds. Add the egg, egg yolks, and vanilla and pulse a few more times, just until the dough starts to come together, about 10 seconds.
Turn the dough out onto a lightly floured work surface and divide it into 4 equal pieces. Shape each piece into a disk 3/4 to 1 inch (2 to 2.5 cm) thick, then wrap each disk in plastic wrap, and refrigerate for 1 hour.
Position 2 oven racks, evenly spaced, in the middle of the oven and preheat the oven to 350°F (177°C). Line 2 rimmed baking sheets with parchment paper.