Preheat the oven to 350 ˚F and grease a 9×13” pan with shortening. Cut a sheet of parchment paper to fit the bottom of the pan and grease that as well.
Cream the butter and sugar in the bowl of a stand mixer with a paddle attachment until light and fluffy. Add in the pudding, salt, and vanilla. Mix until well combined and scrape the bowl as needed.
Add the flour and mix on low until just combined.
Press the mixture as evenly as possible in the prepared pan. If your hands get warm and sticky (mine always do), wet them with cold water and dry.
Bake for 35-45 minutes until the edges just begin to brown. Remove from the oven and immediately use a bench scraper or knife to cut the shortbread into pieces.
Let cool completely in the pan.
Melt the chocolate and shortening in the bowl of a double boiler until smooth. Chop the pistachios while the chocolate is melting.
Dip a third of each cookie in the chocolate and then press into the chopped pistachios. Let dry on a cooling rack with parchment paper underneath to catch drips.
Allow the chocolate to harden before packaging.