In a medium saucepan combine the butter and brown sugar. Heat and stir over low heat until butter is melted. Remove saucepan from heat; stir in vanilla. Cool mixture 15 minutes. Stir in beaten egg, flour, and cocoa powder until combined. Stir in the 3/4 cup pistachio nuts. Divide dough in half. Cover and chill about 30 minutes or until dough is easy to handle.
On a lightly floured surface, roll half of dough at a time to 1/4-inch thickness. Using a 2-inch heart-shape cookie cutter, cut out cookies. Place cookies 1 inch apart on an ungreased cookie sheet.
Bake in a 350 degree F oven about 9 minutes or until edges are firm. Transfer cookies to a wire rack; cool.
In a small heavy saucepan heat and stir chocolate pieces and shortening over low heat until melted. Remove from heat. Dip half of each cookie into chocolate mixture; roll edges of cookie in ground pistachio nuts. Makes about 48 cookies.