In a large bowl, cream 1 cup softened butter with ½ cup powdered sugar using an electric mixer on medium speed until light and fluffy, about 2-3 minutes.
Beat in 2 teaspoons vanilla extract until fully incorporated into the butter mixture.
In a separate medium bowl, whisk together 1¾ cups flour, dry pistachio pudding mix, and ½ teaspoon salt until well combined.
Gradually add dry ingredients to butter mixture, mixing on low speed just until a cohesive dough forms. Do not overmix.
If using, fold in 1 cup chopped pistachios by hand until evenly distributed throughout the dough.
Divide dough in half and shape into balls. Wrap tightly in plastic wrap and refrigerate for at least 1 hour (or up to overnight).
When ready to bake, preheat oven to 350°F and line baking sheets with parchment paper.
Scoop teaspoon-sized portions of dough, roll into balls, and place on prepared sheets about 1 inch apart. Gently press each ball down slightly.
Bake for 8-10 minutes until bottoms are just beginning to turn golden but tops remain pale and soft.
Let cool on baking sheet for 5 minutes, then roll each warm cookie in remaining ½ cup powdered sugar until fully coated.
Transfer to wire rack and cool completely before storing in an airtight container.