Mix the Dry Ingredients – In a medium bowl, whisk together the flour and baking soda. Set aside.
Melt the Butter – Cut the butter into 14 even-sized pieces. In a skillet, melt 10 tablespoons over medium heat for about 2 minutes. Continue cooking, swirling the pan constantly until golden brown and a nutty aroma forms, 1 to 3 minutes. Remove the skillet from heat. Stir in the remaining 4 tablespoons of butter until completely melted.
Mix the Butter and Sugars – In a stand mixer fitted with the whisk attachment, add the melted butter to the mixing bowl. Add the granulated sugar, brown sugar, salt, and vanilla. Whisk on medium speed (setting 5) until combined, 30 seconds.
Whisk and Rest – Add the whole egg and yolk. Whisk on medium-high speed (setting 8) until smooth with no sugar lumps remaining, about 30 seconds. Let the mixture stand for 3 minutes.
Repeat the whisking and resting process 3 more times: 30 seconds mixing at medium-high speed and 3 minutes of rest. After four total times, the mixture should be thick, smooth, and shiny.
Add the Peanut Butter – Whisk in the peanut butter on low speed (setting 2) until just combined, about 30 seconds.
Add the Flour and Mix-ins – Using a rubber spatula or wooden spoon, stir in the flour mixture until just combined, about 1 minute. Stir in 1 ¼ cups of chocolate chips and chopped peanuts, giving the dough a final stir to ensure no flour pockets remain.
Chill the Dough – Cover the mixing bowl with plastic wrap and transfer it to the refrigerator or freezer. Chill until the dough firms up, around 65 to 70ºF (18 to 21ºC), about 15 minutes.
Preheat the Oven – Adjust the oven rack to the middle position. Heat to 350°F (177ºC). Line two large baking sheets with parchment paper.
Roll Out Dough Balls – Divide the dough into 18 portions, each about 3 tablespoons. Firmly press the dough into balls. Arrange at least 2″ apart on lined baking sheets, 9 per tray. Flatten the balls into ¾” discs. Press the sides together to create rounded edges. Use the remaining ¼ cup of chocolate chips to press some on top of the dough.
Bake the Cookies – Bake one tray at a time for 5 minutes, then rotate the baking sheet. Continue baking until the edges are golden brown and the centers are just set, about 5 to 7 minutes.
Let Them Cool – Leave the cookies on the baking sheet for 5 minutes, then transfer them to a wire rack. Cool completely to set the structure. Bake the remaining tray once the oven is back to temperature.