White chocolate: In a small bowl, add the chopped white chocolate and sift in the matcha powder. Set aside.
Brown the butter: In a small saucepan, melt the butter on medium heat. Continue heating, stirring often, until the white milk solids separate out and the butter starts to sizzle. When the sizzling subsides and it starts to foam, keep stirring and watch the milk solids at the bottom of the pan carefully. Once the milk solids become golden brown and the butter smells nutty, remove it from the heat immediately.
Melt chocolate: Pour the browned butter over the white chocolate and let it sit undisturbed for 3 minutes. Stir until the white chocolate has completely melted and the matcha powder is combined. Set aside to cool slightly while you make the rest of the batter.
Prep: Preheat the oven to 350°F. Line a 9x9" square pan with parchment paper on all sides and set aside.
Whip eggs: To a large mixing bowl, add the eggs and sugar. Use an electric hand mixer fitted with a whisk attachment to beat them together for 7-10 minutes at medium speed or until it thickens and becomes very pale in colour.
Stream in white chocolate mixture: With the mixer on low speed, stream the white chocolate mixture into the egg mixture. Mix until combined.
Fold in flour: Add the flour and fold together with a spatula until just combined.
Fill pan: Transfer the batter to your lined pan and use an offset spatula to spread the batter into a flat layer and into the corners of the pan.
Bake: Bake for about 25 minutes or until the edges are set but the middle is still slightly jiggly. Let the brownies cool completely in the pan on a wire rack.
Slice: Lift them out of the pan by grasping the parchment paper overhangs. Once completely cooled, slice the brownies into squares.