Brown the butter. Melt the butter in a light-colored pan over medium heat. Stirring constantly with a spatula or wooden spoon, continue to cook for 5 to 10 minutes, or until the butter smells nutty and the milk solids at the bottom of the pan turn golden brown. The butter will foam around the edges in the first few minutes of cooking. The foam will subside slightly before the butter is fully browned. Immediately pour the browned butter into a large heatproof bowl to prevent burning and set aside to cool slightly.

In a medium bowl, whisk together the flour, baking soda, and salt.

Add the brown sugar and granulated sugar to the bowl with the butter and whisk to combine. Add the egg and egg yolk and whisk until well combined and no streaks of egg white remain. Whisk in the vanilla.

Add the dry ingredients to the wet ingredients and mix with a spatula. When the dry and wet ingredients are mostly combined, add the chocolate chips and mix until well combined.

Bake the cookie dough right away, or cover and refrigerate for up to 2 days.

When ready to bake, preheat the oven to 350°F and line two large baking sheets with parchment paper.

Use a ¼-cup cookie scoop to scoop the dough onto the baking sheets, leaving at least 2 inches between cookies. Press more chocolate chips into the tops of the cookies, then press down lightly on the tops of the dough mounds to flatten them slightly. Bake, one sheet at a time, for 10 to 13 minutes, or until the edges of the cookies are golden brown but the tops still look slightly underdone. The cookies will set up as they cool.

Let cool on the baking sheets for 10 minutes before transferring to a wire rack to cool completely.