In a large saucepan, melt the butter over low heat. Once fully melted, increase the heat to low-medium to cook the butter, stirring frequently with a rubber spatula or wooden spoon. The butter will start popping and crackling and then start foaming. The part of the butter that will start browning is the milk solids which will be on the bottom of the pan. This is why you need to keep stirring and lifting the spatula or spoon out of the pan to see what stage it is at. Once you have nicely browned specks on the bottom of the pan, remove the pan from the heat and immediately pour the butter into a heatproof bowl to stop it from cooking any further. (Be careful not to overcook the butter. It can go from brown to burnt very quickly.) Allow the butter to cool to room temperature. Cover and refrigerate until hardened, at least two hours. (Brown butter can be made a day or two in advance and stored in the refrigerator.)
In a small bowl, combine the flours, baking soda and salt.
In an electric mixer fitted with the paddle attachment, cream together the brown butter, sugar, brown sugar, milk, vanilla and vinegar. Add the egg and beat until fully combined.
Add the dry ingredients and mix just until fully combined. Mix in the chocolate chips.
Divide and shape into rounds, each weighing 30 grams. Refrigerate at least one hour and up to two days.
Preheat the oven to 375°F.
Sprinkle sea salt on top of each cookie and bake the cold cookies 11-13 minutes, just until the edges are light brown and the middle looks a bit doughy.
Cool for a few minutes on pan, then on a wire rack.