Preheat oven to 325F. Line an 8x8-inch metal baking pan with parchment, leaving a 2-inch overhang on two sides for easy lifting. Butter the exposed sides.

In a medium saucepan over low heat, melt the butter with 4 oz of the white chocolate, stirring constantly until smooth. Remove from heat and whisk in both sugars until dissolved and glossy, about 1 minute. Let cool 5 minutes so the eggs don't scramble.

Whisk in eggs one at a time until fully incorporated, then stir in vanilla. Add flour, matcha, and salt. Fold with a spatula until just combined and no streaks remain - do not overmix. Fold in the remaining 2 oz white chocolate chunks.

Scrape batter into the prepared pan and smooth the top. Bake 32-36 minutes, until the top is crackly and set but a toothpick inserted 1 inch from the edge comes out with moist crumbs. The center should still have slight jiggle - it will set as it cools.

Cool completely in the pan on a wire rack, at least 1 hour. Use the parchment overhang to lift out, then cut into 16 squares with a sharp knife, wiping clean between cuts for neat edges.