In a mixer fitted with the paddle attachment, cream the butter, peanut butter, and sugars on medium speed until light and fluffy, about one minute.
Add the egg, half and half, and vanilla extract, and mix for an additional minute. Mix in the flour, chips, baking soda and salt until incorporated. Cover the bowl and place in the refrigerator for one hour.
Preheat oven to 375° F. Line a couple baking trays with aluminum foil or parchment paper. Set aside.
Scoop the dough onto the prepared trays (I use a 1-ounce ice cream scoop). Flatten slightly with palm. Mark the dough with the back of a fork in a criss-cross pattern.
Bake for 9 to 10 minutes, or until edges are just set. Cookies will look underbaked. Remove from oven and let the cookies cool completely on the sheet trays.