Preheat your oven to 350°F (175°C) and line a baking sheet with parchment paper.
Mix the ground flaxseed and water in a small bowl. Set it aside for 5 minutes to thicken.
Add the roughly chopped vegan white chocolate and vegan butter to a heatproof bowl. Microwave in 15-second intervals, stirring frequently until completely smooth. Let it cool slightly.
In a separate medium bowl, sift the flour, cornstarch, baking powder, salt, and matcha powder. Use a whisk to mix them together.
In a large mixing bowl, combine the cooled chocolate and butter mixture with the sugar, thickened flax eggs, almond milk, and vanilla extract. Use a hand mixer on medium speed, to beat the ingredients until the mixture is smooth and fully incorporated. You can also do this by hand with a whisk.
Gradually add the dry ingredients to the wet ingredients in 2 to 3 batches. Use your hand mixer on the lowest speed (or a sturdy rubber spatula) to mix until just combined. Stop immediately once the flour disappears.
Cover and place the bowl in the refrigerator for at least 30 minutes.
Use a cookie scoop to form the chilled dough into balls (about 1.5 tablespoons each). Place them on your lined baking sheet and press them flat with the palm of your hand.
Bake for 10 minutes at 350°F. When ready, they will look done on the edges and a little underdone in the middle.
Let the cookies cool on the hot baking pan for 5 minutes before transferring to a wire cooking rack.
Store baked cookies in an airtight container at room temperature for up to 4 days. You can also freeze the baked cookies for up to 2 months.