First, make the brown butter. Place the unsalted butter in a medium saucepan on the stove over medium-low heat. Heat until the butter is fully melted. Once the butter is melted, stir the butter constantly and continue to cook until it foams up and the milk solids turn a golden brown color. Pour the brown butter into a heat-proof bowl and cool to room temperature. Cool the butter to room temperature and then chill it in the refrigerator to solidify. Before making the cookies, allow the brown butter to come to room temperature.
In the bowl of a stand mixer fitted with a paddle attachment or in a large mixing bowl with an electric hand mixer, cream together the softened brown butter, brown sugar, and granulated sugar until well combined and smooth. Add in the eggs and vanilla extract and mix until well combined.
Add the all-purpose flour, baking soda, and salt. Mix on low speed until the cookie dough forms. Fold in the dark chocolate chunks until well dispersed throughout the cookie dough.
Use a large cookie scoop (3 tablespoons) to portion the cookie dough. Place the cookie dough balls on a parchment-lined baking sheet pan. If desired, press additional chocolate chunks into the dough balls. Chill the cookie dough for 30 minutes.
Preheat the oven to 350℉ (177℃). Line baking sheet pans with silicone baking mats or parchment paper. Place the chilled cookie dough balls on the prepared baking pans, allowing room for spreading. Bake the cookies for 12 to 13 minutes until the cookies are golden-brown on the edges and slightly underbaked in the center.
Remove the baking pan from the oven. Allow the cookies to cool on the pan for about 2 minutes and then transfer the cookies to a wire rack to cool.