Beat the butter, vanilla, and powdered sugar in the bowl of a stand mixer fitted with a paddle attachment on medium speed for about 5 minutes, or until light and creamy. Scrape the bowl a few times during mixing.
Whisk the all-purpose flour and salt together in a separate bowl. Turn the mixer to its lowest speed and gradually add the dry ingredients to the butter mixture.
Use a spatula to fold all of the pistachios into the dough.
Transfer the firm yet pliable dough to a large sheet of parchment paper. Roll the dough into a 10-inch log, about 2 inches thick. Wrap the dough in parchment paper and refrigerate for about 3 hours or overnight.
Once the dough is firm, cut the log into ¼-inch rounds. Place the cookies on a parchment-lined baking sheet, about 1 inch apart. Bake at 350°F for about 13 to 15 minutes, or until lightly golden around the edges.
Let the cookies cool on the baking sheet for a few minutes, then transfer them to a cooling rack to cool completely.
Melt the chocolate in the microwave or in a double boiler.
Dip the cookies into the melted chocolate, then place them on parchment paper to set. If desired, sprinkle the cookies with extra chopped pistachios and coarse salt or salt flakes.