Melt butter in a small heavy saucepan over medium, stirring constantly, until butter begins to turn golden brown, 6 to 8 minutes. Immediately remove saucepan from heat, and pour butter into a small heatproof bowl. Cover and chill until butter is cool and begins to solidify, about 1 hour.
Preheat oven to 350°F. Line 3 baking sheets with parchment paper. Beat browned butter, brown sugar, and granulated sugar in a stand mixer fitted with a paddle attachment on medium speed until creamy, about 1 minute. Add eggs and vanilla, beating until blended, about 30 seconds.
Stir together flour, baking soda, and salt in a small bowl; gradually add to browned butter mixture, beating on low speed, until just blended. Beat in chocolate chips and nuts until just combined.
Using a 2-tablespoon cookie scoop, scoop and drop cookies 2 inches apart on prepared baking sheets. Bake cookies in preheated oven in batches, 1 baking sheet at a time, until cookies are golden and set around the edges, 11 to 13 minutes per batch. Transfer cookies to wire racks; let cool completely, about 15 minutes.