Cut the butter into chunks and melt in a sauce pan over medium heat. I suggest using a pan that has a light color bottom as this will make it easier to see the color of the butter as it browns. Once the butter melts, bring it to a boil. Once it begins to boil, swirl the butter in the pan over the heat constantly for 3-5 minutes, until the butter is no longer very foamy and turns a deep amber color. It will also have a very nutty smell. Transfer the butter to a separate bowl and allow to sit and cool for 15-20 minutes. Putting the bowl over an ice water bath will help with this.
Preheat oven to 350°F. Line baking sheet(s) with parchment paper.
While butter is cooling, whisk together the flour, baking soda, and salt in a separate bowl.
Once the butter has cooled, add the brown sugar, granulated sugar, and vanilla to the butter and beat together until it is combined. The butter may still be warm at this point - warm is fine, but you don't want it hot, otherwise it will scramble the eggs. Beat in the eggs one at a time until the mixture is well combined. You may see little dark flecks in the mixture - this is from browning the butter and is what will give these cookies their extra special taste!
Add the flour mixture to the egg-sugar-butter mixture. If you are using a stand mixture, beat on medium-low until just combined, about 30 seconds. If you are not using a stand mixer, use a wooden spoon to mix by hand until the dough just comes together. You don't want to overmix once the flour is added.
Use a wooden spoon to fold in the chocolate chips. I prefer a mix of half-and-half semi-sweet and dark chocolate for a bit of extra decadence.
Use a spoon or cookie scoop to make drops of cookie dough approximately 2 tablespoons each and drop them onto the lined baking sheet, leaving bout 2 inches between each dough drop.
Bake at 350°F for 10-12 minutes until edges are lightly gold. If you prefer a more gooey chocolate chip cookie, you can bake for 8-9 minutes instead.
Transfer to a wire rack to cool, and sprinkle each cookie with a small amount of flaked or coarse sea salt to finish them off.
Can be stored in an airtight container at room temperature for up to 5 days.