Adjust oven rack to middle position and heat oven to 375 degrees. Line 2 large (18- by 12- inch) baking sheets with parchment paper. Whisk flour and baking soda together in medium bowl; set aside.
Heat 10 tablespoons butter in 10 inch skillet over medium-high heat until melted, about 2 minutes. Continue cooking, swirling pan constantly until butter is dark golden brown and has nutty aroma, 1 to 3 minutes. Remove skillet from heat and, using heatproof spatula, transfer browned butter to large heatproof bowl. Stir remaining 4 tablespoons butter into hot butter until completely melted.
Add both sugars, salt, and vanilla to bowl with butter and whisk until fully incorporated. Add egg and yolk and whisk until mixture is smooth with no sugar lumps remaining, about 30 seconds. Let mixture stand 3 minutes, then whisk for 30 seconds. Repeat process of resting and whisking 2 more times until mixture is thick, smooth and shiny. Using rubber spatula or wooden spoon, stir in flour mixture until just combined, about 1 minute. Stir in chocolate chips and nuts (if using), giving dough final stir to ensure no flour pockets remain.
Divide dough into 16 portions, each about 3 tablespoons (or use #24 cookie scoop). Arrange 2 inches apart on prepared baking sheets, 8 dough balls per sheet. (Smaller baking sheets can be used, but will require 3 batches).
Bake cookies 1 tray at a time until cookies are golden brown and still puffy, and edges have begun to set but centers are still soft, 10 to 14 minutes, rotating baking sheet halfway through baking. Transfer baking sheet to wire rack; cool cookies completely before serving.