In a large bowl, use an electric hand mixer or a stand mixer fixed with the paddle attachment to cream the butter, peanut butter and both sugars until pale and fluffy, about 4 minutes.
Beat in the egg and vanilla extract.
In a medium bowl, combine the flour, baking soda and salt.
By hand with a wooden spoon or with the mixer on low speed, mix the dry ingredients into the wet ingredients.
Stir the chocolate chips into the dough.
Portion out the dough by the 1/4-cup-full onto a sheet pan or a plate lined with parchment paper or wax paper. Cover them and chill overnight in the refrigerator.
Preheat the oven to 350 degrees F. Line 2 baking sheets with parchment paper.
Place the cookie dough balls on the pan 3 inches apart.
Bake the cookies for 20-22 minutes until they are golden brown and cracked on top. (For the most consistent baking, put one pan at a time on the center rack in the oven. Alternate between the two baking sheets, so you do not put cookie dough on a hot pan.)
Cool for 5 minutes on the pan and then transfer to a wire rack to continue cooling.