For the shortbread: Preheat the oven to 325 degrees F. Pulse the pistachios until finely ground in a food processor. Remove.
Combine the butter, confectioners' sugar, lime juice, lime zest, cardamom and salt in the food processor and process until well combined. Add the flour and pistachios and pulse to combine.
Pat the dough into an 8- by 8-inch baking pan and smooth the top to flatten evenly. Pierce the dough all over with a fork. Score the shortbread into 4- by 4-inch pieces using a paring knife, making a total of 16 pieces.
Bake until the cookies are set and golden brown, 25 to 30 minutes. Immediately cut the shortbread through the scored lines; cool in the pan on a wire rack.
For the glaze: Mix together the confectioners' sugar and lime juice in a small bowl. Spread the glaze over the cooled cookies and sprinkle the lime zest over the top.