In a medium saucepan, cook the butter over moderate heat until it is browned and nutty-smelling, about 5 minutes. Transfer the butter to a small bowl and let it cool to room temperature.
In a medium bowl, whisk together the flour, salt and baking soda. In the bowl of a standing mixer fitted with the paddle, beat the browned butter with the granulated and light brown sugars at medium speed until light and fluffy, scraping down the bowl halfway through, about 7 minutes. Beat in the whole egg, egg yolk, milk and vanilla. At low speed, beat in the dry ingredients, scraping down the bowl as needed. Mix in the chopped chocolate. Cover the bowl with plastic wrap and refrigerate overnight.
Preheat the oven to 375°. Line 3 large baking sheets with parchment paper. Scoop 1/4-cup mounds of cookie dough onto each baking sheet. There should be 6 cookies per sheet. Bake one sheet at a time for about 18 minutes, until the cookies are just firm and golden brown on the bottom. Transfer the sheet to a rack to cool. Repeat to bake the remaining cookies. Serve the cookies warm or at room temperature.