Preheat the oven to 375 degrees and grease an 8x8 brownie pan with butter or cooking spray and line the pan with parchment paper to avoid sticking.
Cut the butter into large chunks. Put them in a pan and cook them on medium heat until the butter solids brown and the butter smells nutty and stay close, stirring continuously. This should take about 15 minutes. Pour it into a separate bowl and let it cool for about 15 minutes.
While the butter cools, mix the flour, baking powder, and salt in a bowl. Set aside.
In a stand mixer, add the brown sugar, granulated sugar, and browned butter. Beat on medium speed for 2-3 minutes. Add the eggs, one at a time. Add the vanilla and continue to mix. Be sure to scrape the bowl to get everything off the edges of the bowl. With the mixer on low speed, carefully add in the dry ingredients. Mix until combined and then turnoff the mixer. Add in the peppermint bark, white chocolate chips, and chocolate chips. Mix those in by hand as to not overmix.
Pour the mixture into the prepared pan and cook for 35-40 minutes, depending on your oven. A toothpick should be inserted and come out clean. Let the blondies cool for a few hours before cutting. Serve immediately, or store in an airtight container for 2-3 days once cooled.