Preheat oven to 375°.
In a large bowl from your mixer, combine the cold cubed butter and both sugars. Fitted with the paddle attachment, mix ingredients on a medium speed for about 4 minutes or until creamy.
Add eggs, one at a time. Be sure to mix well after adding each one. Then add vanilla and mix to combine.
In a separate bowl combine flour, cornstarch, baking soda and salt. Whisk ingredients together and then add to the butter and sugar mixture all at once. Mix until just combined. You do not want to over mix {this will heat up your butter}.
Using a medium cookie scoop {or 2-3 tablespoons of dough} get a mound of dough. Add a peanut butter cup to the middle of the scoop and press down into dough. Scoop another mound of dough and place on top of the peanut butter cup. Using your hands bring the two cookie dough mounds together around the peanut butter cup, being sure to seal edges. Place on a pan lined with parchment or wax paper and continue making cookies until all of the dough is gone {should be about 8-10 VERY large cookies}.
Bake for 10-13 minutes, or until the edges are set and the middle of the cookie is just beginning to set and barely brown. They will look under baked but let them cool completely on cookie sheet and they will firm up as they cool.
Once cookies have cooled completely melt peanut butter and chocolate in separate bowls in the microwave. I like to start with 30 seconds, stir and add more time if it need it. Both should be smooth and thin. Using a spoon or fork drizzle both peanut butter and chocolate over the cookies.
Store cookies in an air tight container at room temperature for up to a week or freeze for up to 3 months.