Preheat the oven to 350F degrees.
In a large bowl beat together the butter, peanut butter and sugars using an electric mixer until no lumps remain.
Beat in the eggs and vanilla.
Turn the mixer to low speed and carefully beat in the flour, cornstarch, baking soda and salt.
Form the dough into balls about 1.5-2 tablespoons in size and place at least 2 inches apart on a cookie sheet lined with parchment paper.
Bake for 8-10 minutes or until the tops look just set.
Cool on the cookie sheet for 10 minutes, then transfer to a wire rack to continue cooling.
Melt the chocolate chips and butter in a double boiler or in 30 second bursts in the microwave, stirring between each.
Dip one half of each cookie with melted chocolate then sprinkle with chopped peanuts.