Melt the butter in a 2-quart saucepan over low heat. Set aside to cool slightly.
Sift together the flour, salt and baking soda onto a paper plate. Pour the butter into your mixer bowl. Add the sugar and brown sugar and beat on medium speed for 2 minutes.
Meanwhile, whisk together the whole egg, the egg yolk, milk and vanilla extract in a measuring cup. Reduce the mixer speed and slowly add the egg mixture. Mix until thoroughly combined, about 30 seconds.
Using the paper plate as a slide, gradually integrate the dry ingredients, stopping a couple of times to scrape down the sides of the bowl. Once the flour is worked in, drop the speed to "stir" and add the chocolate chips, peanut butter chips, and walnuts. Chill the dough for 1 hour.
Preheat the oven to 375 degrees F and place racks in the top third and bottom third of the oven.
Scoop the dough into 1 1/2-ounce portions onto parchment-lined half sheet pans, 6 cookies per sheet. Bake 2 sheets at a time for 15 minutes, rotating the pans halfway through. Remove from the oven, slide the parchment with the cookies onto a cooling rack and wait at least 5 minutes before devourin