Place the unsalted butter in a small, light-colored saucepan over medium-low heat. Stir or swirl the pan as the butter melts, foams, and begins to sizzle. Continue cooking until the bubbling becomes quieter, the milk solids turn golden brown, and the butter smells nutty. Remove the pan from the heat as soon as the solids reach an amber color. Pour all the browned butter, including the toasted bits, into a heatproof mixing bowl. Let it cool until warm rather than hot.

Add the light brown sugar and granulated sugar to the warm browned butter. Whisk until the mixture looks evenly moistened and resembles a thick, grainy caramel paste. Add the egg yolk and vanilla extract. Whisk until the mixture becomes smoother, slightly thicker, and glossy. It should look cohesive, though some sugar granules may remain visible.

Sprinkle the all-purpose flour, baking soda, and fine salt over the wet mixture. Fold with a silicone spatula until no dry flour remains. The dough should be thick, soft, and slightly tacky rather than runny. If it looks greasy or unusually loose, allow it to cool further before adding the chocolate. Avoid prolonged mixing once the flour is incorporated.

Add the 1/3 cup (57 g) portion of semisweet chocolate chips and fold gently until distributed throughout the dough. Reserve the remaining 2 tablespoons plus 2 teaspoons (28 g) for topping. The dough should hold together when pressed and should not leave a pool of melted butter at the bottom of the bowl.

Divide the dough into 6 equal portions and shape them into tall, rounded mounds rather than flattening them into discs. Place the portions on a parchment-lined baking sheet, leaving room for spreading. Press the reserved chocolate chips onto the tops. Refrigerate for 30 minutes. The dough should feel firmer and less sticky after chilling.

While the dough chills, preheat the oven to 350°F (175°C). Position a rack in the center of the oven. Bake the chilled cookies on the center rack for approximately 12 minutes, until the edges are lightly golden and set but the centers still look slightly soft. The centers may appear a little underdone when the cookies first come out of the oven. They should not look wet or liquid. If the cookies are still glossy and raw-looking across the entire surface, continue baking briefly and check again.

Let the cookies rest on the baking sheet for 10 minutes. During this time, the centers will firm up and the edges will become more stable. Transfer carefully to a wire rack. Serve while slightly warm for softer chocolate or allow them to cool completely for a more structured, chewy texture.