Place 1 stick plus 1 tablespoon of butter in a small pan over medium heat. Once the butter melts it will begin to foam and froth and will make a crackling sound as the water cooks out of it. You can swirl the pan from time to time but don't really need to stir it. Once it stops crackling keep an eye on it. The butter will begin to brown and smell nutty. Once it is a nice chestnut color remove it from the heat and pour it into a bowl or other small container, making sure to use all the brown bits in the bottom. Let cool slightly.
In the bowl of a stand mixer fitted with the paddle attachment, cream the remaining 1 stick of butter with the granulated sugar on medium speed until light and fluffy, 3 to 5 minutes.
Add the vanilla extract and molasses and beat until incorporated. Once the browned butter has cooled, pour it into the creamed butter mixture along with the brown sugar. Cream for about 2 minutes until well incorporated. Add the egg and egg yolk and beat another minute.
In a medium bowl, whisk together the flour, salt, and baking soda. Add it all at once to the butter mixture and beat on low speed until the flour is just incorporated. Remove the bowl from the stand mixer and use a spatula or wooden spoon to fold in the walnuts and chocolate chips.
Cover the bowl with plastic wrap and place in the refrigerator for at least 30 minutes.
Preheat the oven to 375ºF. Line a baking sheet with parchment paper or a silpat. Place tablespoons of dough about 2 inches apart on the prepared sheet. Bake for 12 to 14 minutes until lightly browned but still soft in the middle. Remove from the oven and let rest on the sheet for 5 minutes before removing to a cooling rack to cool completely.