Preheat the oven to 325 degrees F. Line the baking sheet with parchment paper or a silicone baking mat.
Add room temperature egg whites to a medium bowl. Add the sugar and cream of tartar to the egg whites.
Beat the egg whites with an electric mixer until soft peaks form about 3 to 5 minutes.
Once the egg whites are ready, add the vanilla and beat in.
Add the chopped pistachios and coconut to the egg whites and fold in until well combined.
With an ice cream scoop or a tablespoon drop mound of the coconut mixture on the prepared baking sheet. Shape the mounds if they need shaping.
Place the cookies into the preheated oven and bake 18 to 20 minutes or until the cookies are lightly browned.
Remove from the oven and place the cookies on a cooling rack and cool completely
Once the cookies are cooled. Place the white chocolate chips into a microwave-safe bowl with the shortening and melt. Heat for 20 seconds at a time. Mixing the chocolate with a spoon in between each 20 seconds.
Drizzle the melted white chocolate over the cookies. Let the chocolate dry and store in an airtight container.