Place the butter in a small saucepan over medium heat. Melt completely, then cook while stirring often until the butter foams and brown specks form at the bottom with a nutty aroma, about 8–10 minutes. Immediately pour into a large bowl and cool for 20–30 minutes until slightly warm.
In a medium bowl whisk together the flour, baking soda, and salt until evenly combined.
Add the brown sugar and granulated sugar to the cooled brown butter. Mix with a spatula until thick and slightly grainy.
Add the egg, milk, and vanilla extract. Stir gently until the mixture becomes smooth and creamy.
Add the flour mixture to the bowl and fold slowly until almost combined.
Add the chopped chocolate and chocolate chips. Continue mixing just until the dough comes together and the chocolate is evenly distributed.
Cover the bowl and refrigerate the dough for 1 to 4 hours to allow the flour to absorb moisture and help control spreading.
Preheat the oven to 375°F (190°C) and line two baking sheets with parchment paper.
Scoop the dough into about 14 portions (around 2 oz or 56 g each). Roll into smooth balls and place them on the baking sheets about 3 inches apart. Add extra chocolate pieces on top if desired.
Bake for 8–10 minutes until the edges are deep golden and the centers look slightly soft. Let the cookies rest on the baking sheet for 2 minutes before transferring to a wire rack. Sprinkle with flaky sea salt if desired.