Preheat oven to 350ºF. Lightly grease a metal 8x8-inch baking pan and line with a piece of parchment that slightly hangs over the side. Set aside.
Brown the butter. Add the butter to a small saucepan. Cook, swirling the butter occasionally, until golden brown with a nutty aroma. Pour the brown butter into a heat-safe mixing bowl and cool for 10 minutes.
Mix in sugar and eggs. Whisk both sugars into the butter until very smooth, about 2 minutes of whisking. Whisk in the egg, egg yolk, and vanilla until smooth.
Mix in the dry ingredients. In a separate bowl, whisk together the flour, baking powder, baking soda, malted milk powder, and kosher salt. Use a rubber spatula to fold the dry ingredients and chopped chocolate into the wet until mixed and no streaks of flour remain.
Bake. Press the cookie dough into the prepared pan and use a small offset spatula to spread the cookie dough into a smooth, even layer. Bake for 22-27 minutes, until the edges are slightly puffed and the center is just barely set (a little gooey is good!).
Salt and cool. Remove from the oven and sprinkle with flaky sea salt. Let the cookie cool for 45 minutes before removing from the pan and cutting into bars.