Preheat oven to 325 degrees F. Grease a 9-inch square baking pan and line with parchment paper; set aside.
Place the butter and chopped chocolate in a heatproof bowl set over a pan of barely simmering water. Stir occasionally until completely smooth and melted. Remove the bowl from the heat and allow to cool slightly.
Meanwhile, in a medium bowl, sift together the flour, baking powder and salt.
In a large bowl, whisk together the sugars, eggs and vanilla until completely smooth and there are no lumps of sugar. Add the melted chocolate mixture and whisk gently until thoroughly combined.
Add the flour mixture, white chocolate chips, milk chocolate chips and dark chocolate chips and, using a rubber spatula, fold into the chocolate batter until no streaks of flour remain.
Pour the batter into the prepared pan, smoothing it into an even layer. Bake for 45 minutes to 1 hour, until a crust has formed on top and a toothpick inserted into the center comes out covered in sticky crumbs.
Allow the brownies to cool completely in the pan, then slice into squares and serve. They can be stored at room temperature in an airtight container for up to 4 days, or wrapped individually and frozen for up to 1 month.