Melt butter in a small saucepan over medium-low. Cook, stirring occasionally, until butter is foamy and milk solids at the bottom of saucepan turn brown, about 5 minutes. Immediately pour into a small heatproof bowl; cover and refrigerate until solidified, 45 minutes to 1 hour.

Meanwhile, preheat oven to 350°F. Sift together flour, baking powder, kosher salt, and baking soda in a medium bowl and set aside.

Beat chilled browned butter, brown sugar, and granulated sugar with an electric mixer (if working with a stand mixer, use the paddle attachment) on medium-high speed until light and fluffy, about 3 minutes. Add egg and vanilla; beat on medium speed until well combined, about 15 seconds. With mixer running on low speed, gradually add flour mixture, beating just until combined, about 1 minute. Fold in white chocolate, pistachios, and ¼ cup of the dark chocolate until evenly distributed. Portion dough into 24 pieces, about 1 tablespoon each; roll dough into 1-inch balls and place in a 24-cup non-stick mini muffin tray.

Bake in preheated oven until edges are set, but centers are still slightly soft, 12 to 14 minutes. Using a rounded 1-teaspoon measuring spoon, press down center of each cookie to create a well. Let cool in muffin tray on a wire rack for 15 minutes. Loosen edges with a knife and carefully remove cookie cups from muffin tray. Let cool completely on wire rack, 20 to 30 minutes.

Microwave remaining ¼ cup dark chocolate in a small microwave-safe bowl on HIGH until chocolate is mostly melted, 45 to 55 seconds, and stir until chocolate is completely melted. Set aside. Spoon pistachio cream into cooled cookie cups, about 1 teaspoon each. Drizzle melted chocolate over filled cookie cups. Sprinkle with pistachios. Refrigerate, uncovered, until chocolate is set, about 5 minutes. Store in an airtight container at room temperature for up to 1 week.