Preheat the oven to 375°F. if you are planning to bake the cookies right away, otherwise preheat the oven 30 minutes before you bake them.
In a large bowl, whisk the flour, salt, and baking soda together and set aside.
Using an electric mixer, beat the butter, brown sugar, and granulated sugar together in a large bowl until creamy and well blended. Scrape down the bowl and add the eggs, one at a time, beating until each is incorporated. Blend in the vanilla.
Gradually add the flour mixture, mixing each addition in until incorporated. Blend in the chocolate pieces and peanuts.
If not baking immediately, press plastic wrap directly onto the dough in the bowl and put in the refrigerator. If you can refrigerate the dough for 1 to 2 days, up to 4 days (the longer the dough sits, the deeper and more developed the flavors become).
Bring the dough to room temperature. Place 1½-inch balls of the dough onto a cookie sheet about 2½ inches apart, and press down lightly, especially if still a bit chilled from the fridge. Bake in the preheated oven for 9 to 11 minutes until nicely browned, but still soft in the middle. Peek part way through the baking time and rotate the trays if the cookies seem to be cooking unevenly.
Remove the pan from the oven and let the cookies sit on the baking sheets for about 1 minute more to firm up a bit. Transfer the cookies onto a wire rack, and either serve warm or cool completely.