Brown the butter: Add the butter to a light-colored pan over medium heat. Stir frequently as the butter melts. It will become foamy, then the milk solids will separate and begin turning golden brown. Continue stirring until the butter smells nutty and turns a deep golden brown. Remove from the heat immediately and transfer to a bowl to prevent burning.
Let the brown butter cool until it is warm to the touch, but not hot.
Add the granulated sugar and brown sugar to the brown butter. Whisk until well combined. Add the egg and vanilla extract and whisk until smooth.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet mixture and gently mix until just combined. Do not overmix.
Fold in the chocolate chips until evenly distributed throughout the dough.
Line a baking sheet with parchment paper. Scoop the dough into 2-tablespoon portions and place them several inches apart.
Bake at 350°F (175°C) for 10–12 minutes, until the edges are lightly golden and the centers still look slightly soft.
For the softest, gooiest cookie, let the cookies cool for just 1–2 minutes, then place a scoop of ice cream on top and serve immediately. I used coffee affogato ice cream.