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1. In a large bowl beat butter with an electric mixer on medium to high for 30 seconds. Add sugar, vanilla, and salt. Beat until combined, scraping sides of bowl occasionally. Split vanilla bean in half lengthwise. Using the tip of a knife, scrape seeds from vanilla bean halves into butter mixture. Beat until combined, scraping sides of bowl occasionally. Beat in as much of the flour as you can with the mixer. Using a wooden spoon, stir in any remaining flour. Stir in pistachios.
2. Divide dough in half. Shape each half into a 1 1/2-inch-thick log (about 7 inches long). Wrap each log in waxed paper; twist the ends. Chill for at least 2 hours or up to 2 days.
3. Preheat oven to 375 degrees F. Line cookie sheets with parchment paper; set aside. Cut chilled or frozen logs into 1/4-inch slices. Place slices about 1 inch apart on prepared cookie sheets. Bake about 10 minutes or just until firm and browned on the bottoms. Cool on cookie sheets for 1 minute. Transfer to wire racks; cool.
4. In a heavy small saucepan combine white chocolate and shortening; heat and stir over low heat until melted. Dip half of each cookie into melted chocolate mixture; place on a wire rack over waxed paper and let stand until set.