Brown the butter: In a medium saucepan, melt butter over medium heat. Once melted, swirl the pot a few times and cook for several more minutes, until mixture starts to get foamy and becomes a deep amber color. Remove from heat and transfer to a heat-proof bowl. Chill until solid, 2-3 hours.
In the bowl of a stand mixer, add butter, sugars, and vanilla. Mix on medium speed until combined, 2-3 minutes. Turn off mixer, scrape down sides of bowl, and mix again for 2 minutes. Mixture should be light and fluffy.
Add the eggs, one at a time, and beat for an additional 2 minutes until well incorporated.
Add flour, baking soda, and salt. Mix on low speed at first, and when flour is almost absorbed, mix on medium until just combined.
Bake for 10-12 minutes, until the edges become a light golden brown.
Let cookies rest on baking sheet for 5 minutes before transferring them to a cooling rack. Store in an airtight container for up to 5 days or freeze for up to 3 months.