Preheat your oven to 350°F and line two baking sheets with parchment paper. Place the butter in a light-colored skillet over medium heat, swirling occasionally until it melts, foams, and develops golden brown bits at the bottom with a nutty aroma, about 5-7 minutes. Immediately pour into a heatproof bowl to stop the cooking and let cool for 15 minutes.
In a large mixing bowl, combine the cooled brown butter, granulated sugar, and brown sugar. Beat with an electric mixer on medium speed until well combined and slightly creamy, about 2 minutes. Add the eggs one at a time, beating well after each addition, then mix in the vanilla extract.
In a separate medium bowl, whisk together the flour, baking soda, and kosher salt. Gradually add the dry ingredients to the wet ingredients, mixing on low speed just until no flour streaks remain. Gently fold in the chopped pistachios and white chocolate chips with a spatula until evenly distributed.
Scoop 2 tablespoons of dough per cookie and roll into balls, placing them 2 inches apart on the prepared baking sheets. Flatten each ball slightly with your palm and sprinkle the tops lightly with coarse sea salt.
Bake for 10-12 minutes, or until the edges are golden brown and the centers look slightly underdone. The cookies will appear soft but will firm up as they cool. Let them rest on the baking sheet for 5 minutes before transferring to a wire rack to cool completely.