Preheat the oven to 350 degrees F. Butter a 13-by-9-inch metal baking pan, then line the bottom and sides with parchment paper; butter the parchment paper. Dust the inside of the pan with flour and knock out the excess.
In a large microwavable bowl, microwave the butter until it melts, about 1 1/2 minutes. Add two-thirds of the white chocolate, then shake the bowl so that the butter covers the chocolate; let stand for 1 minute. Starting from the center of the bowl, whisk the butter and white chocolate together slowly until smooth. Add the matcha powder and whisk until smooth.
In another large bowl, vigorously whisk together both sugars, the vanilla, salt and eggs until smooth, lightened in color and foamy, about 30 seconds. Scrape the matcha mixture into the bowl and whisk until smooth. Add the flour and use a rubber spatula to fold it in until just incorporated.
Scrape the batter into the prepared baking pan, spreading it evenly, and smooth the top. Bake until light brown on top and a toothpick inserted halfway between the center and edges comes out clean, about 35 minutes.
Remove the baking pan from the oven and immediately sprinkle the remaining 4 ounces white chocolate evenly over the top of the "matchies"; let stand for 1 minute. Using a small butter knife or offset spatula, spread the chocolate evenly over the top, creating swirls and peaks.
Let the "matchies" cool completely, then use the parchment paper to lift them from the pan to a cutting board. Slice them into 24 squares and serve. Store leftovers in a single layer in an airtight container for up to 5 days.
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