Melt butter in a small saucepan over medium-low.
Cook and stir occasionally until the butter is foamy. The milk solids at the bottom of the saucepan should turn brown. This will take 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 the chilled browned butter, brown sugar, and granulated sugar with an electric mixer. If you use a stand mixer, use the paddle attachment. Mix at medium to high speed until the mixture is light and fluffy, which takes about 3 minutes.
Add egg and vanilla; beat on medium speed until well combined, about 15 seconds.
With the mixer on low speed, slowly add the flour mixture. Beat it until combined, which takes 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 the preheated oven until the edges are set but the centers are still slightly soft, 12 to 14 minutes.
Using a rounded 1-teaspoon measuring spoon, press down the center of each cookie to create a well.
Let cool in a muffin tray on a wire rack for 15 minutes. Loosen the edges with a knife and carefully remove the cookie cups from the muffin tray. Let cool completely on the wire rack, 20 to 30 minutes.
Microwave the remaining ¼ cup of dark chocolate in a small bowl. Use HIGH power for 45 to 55 seconds. Stir until it is completely melted. Set aside.
Spoon pistachio cream into cooled cookie cups, about 1 teaspoon each. Drizzle melted chocolate over the filled cookie cups. Sprinkle with pistachios.
Refrigerate, uncovered, until the chocolate is set, about 5 minutes.