Brown the butter according to my instructions below this recipe. Once browned, immediately remove from heat and pour into a large tupperware or a 9x13 baking pan. Try to leave some of the browned solids/bits in the pan, unless you like a slightly more complex browned, almost burnt-like flavor. I like to leave them behind and discard. Cover tightly, place in the refrigerator, and chill until solid - about 3 hours.
Remove solid brown butter from the refrigerator and spoon into a large bowl (or the bowl of your stand mixer). Using a hand-held mixer or stand mixer fitted with a paddle attachment, beat the chilled brown butter for 1 minute on medium speed until completely smooth and creamy. Add the granulated sugar and brown sugar and beat on medium high speed until fluffy and light in color. Beat in egg, egg yolk, and vanilla on high speed. Scrape down the sides and bottom of the bowl as needed.
In a separate bowl, whisk the flour, cornstarch, baking soda and salt together until combined. On low speed, slowly mix into the wet ingredients until combined, then beat in the milk on medium speed. The cookie dough will be quite thick. Add the chocolate chips and mix on low for about 5-10 seconds until evenly disbursed. Cover dough tightly with aluminum foil or plastic wrap and chill for at least 3-4 hours1 and up to 3 days. Chilling is mandatory for this cookie dough. The longer the better; I always chill for at least 4 hours.
Remove cookie dough from the refrigerator and allow to sit at room temperature for 20 minutes-- if the cookie dough chilled longer than 4 hours, let it sit at room temperature for about 30 minutes. This makes the cookie dough easier to scoop and roll. Preheat oven to 350F. Line two large baking sheets with parchment paper or silicone baking mats. (Always recommended for cookies.) Set aside.
Once chilled, the dough may be slightly crumbly, but will come together if you work the dough with your hands as you roll into individual balls. Scoop and roll balls of dough, about 1.5 Tablespoons of dough each, into balls. Bake the cookies in batches for 11-12 minutes, until slightly golden brown around the edges. My oven has hot spots and yours may too- so, be sure to rotate the pan once during bake time. The baked cookies will look extremely soft in the centers when you remove them from the oven. Allow to cool for 5 minutes on the cookie sheet. If the cookies are too puffy, try gently pressing down on them with the back of a spoon. During this time, you can press a few extra chocolate chips into the top of the warm cookies-- for looks. The cookies will slightly deflate as you let them cool. After 5 minutes, transfer to cooling rack to cool completely.
Make ahead tip: Cookies stay fresh covered at room temperature for up to 1 week. Baked cookies freeze well - up to three months. Unbaked cookie dough balls freeze well - up to three months. Bake frozen cookie dough balls for an extra minute, no need to thaw.
You may chill the dough using this alternative: chill the cookie dough as a whole for 1 hour. Then remove from the refrigerator and roll into 1.5 Tablespoon balls, as the recipe suggests before baking. Continue to chill the cookie dough balls for the remaining 2-3 hours (or freeze them as suggested in the last step). Some readers find this easier! If you do this, you do not have to let the cookie dough sit at room temperature in the next step.