Preheat the oven to 350°F. Line two baking sheets with parchment paper or silicone baking mats.
Brown the butter: Melt the butter in a light-colored saucepan over medium heat, stirring, until it foams and turns deep golden with brown bits on the bottom. Remove from heat and cool 10–15 minutes.
Mix in the sugars: Using a hand mixer, beat the brown butter with the brown sugar and granulated sugar until smooth and glossy.
Add the egg, egg yolk, milk, and vanilla. Beat again until the mixture is creamy and well combined.
Add the dry ingredients (flour, salt, baking soda, and baking powder) and mix on low or fold with a spatula just until a soft dough forms. Do not overmix.
Fold in about 3/4 cup of the chopped chocolate. If the dough seems dry, add a tiny splash of milk and stir until it comes together.
Scoop dough into balls. Dip the tops of each ball into the remaining chopped chocolate and press gently so it adheres. Slightly flatten if needed. Arrange 6 cookies per sheet. You should have about 22 cookies total.
Bake 10–11 minutes, until the edges are light golden and the centers still look soft. If desired, use the rim of a glass to gently scoot the edges into neat circles right after baking.
Cool on the baking sheet for 2–3 minutes, transfer to a rack, and finish with a sprinkle of flaky sea salt.