In a large mixing bowl, beat the butter and peanut butter on medium speed until creamy. Add 1/2 cup of powdered sugar and beat until light and fluffy. Scrape down the sides of the mixing bowl and blend again until incorporated. Add the vanilla extract and gently mix on low until incorporated, scraping down sides with a spatula as needed.
In a small mixing bowl, whisk together the flour, cornstarch and salt. Add the flour mixture in three batches to the creamed butter mixture. Blend on medium-low until each addition is incorporated. Scrape the dough into a ball, cover the bowl with plastic wrap and refrigerate for 1 hour.
To bake the cookies, preheat oven to 350F. Line a baking sheet with parchment paper. Scoop a level spoonful of dough using a 1-inch cookie scoop. Roll each into a tight ball and place two inches apart on the prepared baking sheet. Press each dough ball gently with the flat bottom of a small glass or measuring cup, to the desired thickness. Bake for 11 to 13 minutes at 350F or until lightly browned around the edges. Slide the parchment paper with cookies onto a wire rack to cool. Repeat with a new piece of parchment paper placed on a cooled cookie sheet. Cool cookies completely before icing.