Preheat the oven to 350°F (180°C). Add the vegan butter and ¼ cup of the chocolate chips to a large, microwave-safe bowl. Heat it in the microwave for a minute. Stir this together into a smooth mixture. Stir in the cane sugar and then let this cool for about 5 minutes.
Meanwhile, make the egg replacer in a small bowl by mixing it with the plant milk until it's lump-free. Let it sit for 3-4 minutes to thicken a bit.
Use a sifter to add the flour, powdered sugar, and baking powder to a separate bowl. Then, whisk them together.
Prep an 8x8 baking dish by greasing it and/or lining it with parchment paper.
Mix the egg replacer into the wet batter. Use a sifter to add the matcha powder. Whisk it into a smooth, green mixture.
Gently stir in the dry ingredients. Then, fold in the remaining white chocolate chips. Spread the batter into a flat layer in the dish and then top it with a few more chocolate chips.
Cover it with foil and bake it for 25-30 minutes. Uncover it during the last 5 minutes and remove it from the oven as soon as you see that the edges are starting to firm up and the center doesn't jiggle when you shake the pan. It will continue to cook and set while it's cooling.