Line 2 baking sheets (half sheet pans) with parchment paper; stack together, then transfer to refrigerator to chill.
Whisk flours and baking soda together in medium bowl; set aside.
Heat 10 tablespoons (145g) butter in 10-inch skillet over medium-high heat until melted, about 2 minutes. Continue cooking, swirling pan constantly until butter just barely starts to turn golden brown, about 1 ½ minutes. Add walnuts and continue cooking, swilling pan, until butter is dark golden brown and has nutty aroma, 2 ½–3 ½ minutes more.
Remove skillet from heat and transfer browned butter to large heatproof bowl with remaining 4 tablespoons butter; using a rubber spatula to scrap butter and solids into bowl. Stir 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 (for a total of 4 mixes) 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 until evenly dispersed.
Divide dough using a #24 cookie scoop, or 3 tablespoon measure, into 20 portions. Arrange dough portions closely on a small baking sheet and freeze for 30 minutes.
Adjust oven rack to middle position and heat oven to 350ºF (176ºC) degrees.
Arrange 6 chilled dough balls 3 inches apart on a chilled parchment-lined baking sheet. (Smaller baking sheets can be used, but will require multiple batches.)
Transfer sheet to oven and bake until cookies are puffed, 10 minutes. Lift baking sheet up about 4 inches, then drop baking sheet down onto rack. Bake until cookies puff again, 2 minutes, then repeat lifting and dropping pan. Bake cookies another 2 minutes, then repeat lifting and dropping pan. Bake cookies until edges are golden brown and centers are slightly puffed and look under-baked, 1–2 minutes more. (Total bake time should be 14–16 minutes.)
Transfer baking sheet to a wire rack; allow cookies to cool completely on sheet before transferring to an airtight container.