For the cookies: Melt 1 1/2 sticks (12 tablespoons) of the butter in a small skillet over medium heat. Cook, swirling the skillet, until the butter is golden brown. Set aside to cool.
Preheat the oven to 325 degrees F. Line 2 baking sheets with parchment paper.
In the bowl of an electric mixer, cream together the brown sugar, granulated sugar and remaining 1/2 stick (4 tablespoons) butter. Drizzle in the cooled brown butter. Mix in the eggs one at a time. Add the vanilla extract.
In a separate small bowl, stir together the flour, baking soda and salt. Add half of the flour mixture to the butter and sugar mixture and mix until incorporated. Add the remaining flour mixture and mix well. Add the nuts and white chocolate and mix until incorporated. Drop large scoops of dough onto the prepared baking sheets using a 2 1/4-inch ice cream scoop.
For the topping: Top the cookies with a few chocolate chunks and chopped nuts. Bake until the edges start to brown, about 14 minutes. Sprinkle the tops with a small amount of sea salt. Let cool on the baking sheets.