In a bowl, whisk together the melted butter with the granulated sugar and brown sugar, until well combined. Then whisk in the vanilla extract and the egg, until smooth.
In a separate bowl, combine the flour, corn starch, salt and baking soda, until everything is evenly distributed.
Using a small food processor or Ninja (or just a sharp chef's knife), chop the pistachios and apricots into small pieces no bigger than 1/4 inch.
Add the flour mixture, chopped pistachios and apricots to the wet ingredients, and stir everything together with a spatula until you have a soft dough.
Use a medium cookie scoop (about 1 1/2 - 2 tbsp capacity) to scoop 18-20 portions of dough. Don't roll the dough balls smooth with your hands – you want the surface to be a little rough and craggy. Just flatten the tops of the dough balls with your fingers.
Cover the dough balls with plastic wrap, and chill for 25 minutes while you preheat the oven.
Preheat the oven to 350 F, and position a rack in the center of the oven. Line a baking sheet with parchment paper.
Space the chilled cookies several inches apart on the baking sheet, baking half at a time, and keeping the rest refrigerated until needed. Bake the cookies for about 10-12 minutes, until the edges are set and light golden brown, and the centers of the cookies are no longer gooey.
Cool for several minutes on the baking sheet, then transfer to a cooling rack to cool completely before icing.