Position two oven racks towards the center of the oven. Preheat oven to 375 degrees F.
Melt the butter in medium sauce pan over medium heat. Continue cooking, for several minutes more (about 3-5), swirling the pan or stirring the butter frequently, until butter is a light golden brown with nutty aroma. Remove pan from heat.
Immediately transfer brown butter to heatproof bowl. Set aside to cool for 15 minutes.
In a small bowl, combine flour and baking soda, stir well. Set aside.
After the brown butter has cooled, add granulated sugar and brown sugar. Stir well until thoroughly combined.
Add salt, vanilla, and eggs to the butter-sugar mixture. Stir well until mixture is smooth.
Pour the flour mixture into the butter-sugar mixture. Stir until the flour is just nearly incorporated.
Add chocolate chunks and walnuts. Stir again until flour is entirely incorporated, and chocolate and nuts are evenly distributed in the dough.
Rest the dough in the refrigerator for 15 minutes.
Form the dough into 16 dough balls, about 3 Tablespoons of dough per cookie. (The raw balls of dough should hold their shape. Chill for several minutes longer if needed. To form the dough balls, I use a 3 Tablespoon scoop, or ¼ cup measuring cup filled about ¾ of the way, for each cookie.)
Place 8 dough balls about 2 inches apart on each of 2 baking sheets. If needed, use your hands to shape each portion of dough into round ball.
Put both baking sheets in oven and bake for 7 minutes, then rotate the position of the baking sheets and bake for another 7 minutes or until cookies are golden brown, slightly darker brown and set around the edges but soft-looking in the center. If you prefer, bake the cookies one sheet at a time in center of oven for 14 minutes each.
Use a large sturdy spatula to transfer cookies to cooling rack.