Whisk together the all-purpose flour (start with 3 cups), sifted matcha powder, baking soda, and salt in a medium bowl until well combined. Set aside.
Cut the butter into cubes and place into a large, heat-safe bowl. Gently heat the butter in 30-second increments in the microwave or on the stove until just melted (you do not want it to be hot).
To the melted butter, add the brown sugar and granulated sugar and whisk for 1-2 minutes until well combined, then beat in the eggs and vanilla extract.
Once the mixture is smooth, fold in all of the dry ingredients. Just before the flour is fully combined, add in the chopped chocolate and mix until there are no more streaks of flour left. If the dough is still too wet, add more flour as necessary—you'll know it's done when you can pinch a bit of dough between your fingers without any sticky residue left behind.
Scoop the dough onto a baking sheet lined with wax or parchment paper (no need to space them apart). Cover the tray with plastic wrap and refrigerate for at least 2 hours but up to 4-5 days. You can bake the cookies without chilling them, but they will spread more and have less flavor—if you're short on time, I recommend freezing the cookies for at least 1 hour before baking.
At this point, the cookie dough can be frozen for later.2
When ready to bake, preheat the oven to 375℉ and line a baking sheet with parchment paper.
Evenly space the cookie dough balls onto the baking sheet with a 2'' gap between each cookie. If desired, press the top of the dough with more chopped chocolate pieces.
Bake for 12 minutes, rotating the tray halfway through. Remove the cookies from the oven once the edges are golden brown and set, but do not overbake. The inside of the cookies should still be slightly raw, as they will continue to cook as they cool. Once out of the oven, sprinkle the cookies with flaky salt (if desired) and allow to cool before serving.