Preheat the oven to 350°F (175°C) and line a 9x9-inch baking pan with parchment paper.
Brown the butter in a saucepan until golden and nutty, then let it cool slightly.
Pat the strawberries dry and toss them with 1 tablespoon of the flour.
Whisk together the remaining flour, baking powder, salt, and cornstarch.
Whisk the brown butter with the brown sugar, then mix in the eggs and vanilla.
Fold the dry ingredients into the wet ingredients until almost combined.
Fold in the strawberries and three-quarters of the chopped white chocolate.
Spread the batter into the prepared pan and sprinkle the remaining chocolate over the top.
Bake for 30-35 minutes until the edges are golden and the center is just set.
Cool completely before slicing into 16 squares.