In a food processor, pulse your pistachios into a fine meal. Transfer to a small bowl. Stir ¼ cup of the sugar and the cardamom into the ground pistachios, and whisk together until well incorporated.
Once the puff pastry has thawed, spread the ½ cup of sugar onto a clean work surface. Carefully unfold the sheet of puff pastry onto the sugared surface. Flip the puff pastry over, so both sides are coated in the sugar.
Using a rolling pin, roll out the sheet of puff pastry into a 10-by-12-inch rectangle. Cut in half so that you have two 10-by-6-inch rectangles. Divide the pistachio cardamom sugar mixture between the two rolled pastries. Spread and gently press into the puff, ensuring the top surface is covered.
Starting with one of the rectangles of pastry. Using your fingers, carefully roll one side of the pastry inward to the center. Then roll the opposite side of the pastry towards the center to meet it. Repeat with the other rectangle of pastry. Place both onto a sheet tray and in the refrigerator to firm back up for about 20 minutes.
In the meantime, preheat your oven to 375° F. Once the pastry has firmed, place it onto a cutting board. Trim the edges of both pastries. Slice into ½-inch thick pieces. Arrange cut side down, about 2-inches apart, onto a parchment-lined baking sheet. (You will need two sheet trays total.)
Bake for about 8-10 minutes or until golden brown on one side. Remove from oven. Using a spatula, carefully flip over each palmier. Place back into the oven and bake for 3-5 minutes more or until golden.
Let cool before removing the palmiers from the trays. Store in an air-tight container for up to 2 days.