Preheat the oven to 350°F (180°C). Line 2 cookie sheets with baker’s parchment paper.
In a large bowl with an electric mixer, beat together the butter and icing sugar until creamy. Add the flour and the salt in 2 or 3 portions, beating well after each addition until incorporated. Add 1 cup (250 mL) of the chopped pistachios, mixing just until combined into a uniform dough. (Reserve the remaining chopped pistachios for later.)
Working with about 1 tbsp (15 mL) of dough at a time, roll the dough between your hands into a 3½ in (8 cm) rope, then bend into a crescent shape and place on the prepared cookie sheet, leaving about 1 in (2 cm) between cookies. Place in the oven and bake for 10 to 12 minutes or until just slightly browned on the bottom (the tops should remain light). Cool on the cookie sheet for about 5 minutes, then remove to a rack to cool completely. Repeat with the remaining dough.
When cookies are cool, place the chocolate and the vege-table shortening in a small microwaveable bowl. Microwave in 30-second bursts until the chocolate is smooth and melted (stir after each 30-second zap). Working carefully so these fragile cookies stay intact, dip ends of each crescent in the melted chocolate and place on a waxed-paper-lined tray. (For a tidier appearance, you may want to scrape the excess chocolate off the bottom before setting each cookie on the tray.) Sprinkle with the remaining chopped pistachios while the chocolate is still soft. Let cool until the chocolate has hardened before removing from the tray and storing in a covered container.