Cream together butter, peanut butter and sugars until smooth. Beat in egg and vanilla.
In a separate bowl, stir together flour, baking powder, baking soda, and salt. Gently stir the dry ingredients into the butter mixture until just combined. (No streaks of flour should be visable but don't over-mix!)
Chill dough in the fridge for at least 45 minutes.
Preheat oven to 375F. Use a small cookie scoop to portion dough into balls and then smush them into more of an oval shape. Place on a parchment-lined cookie sheet 2 inches apart for about 10 minutes or until cookies begin to turn golden. Be careful not to over-bake!
For the Frosting:
Beat the butter with the powdered sugar (Mixture may be dry.) Add the vanilla and milk, gradually until frosting is smooth - scoop about a 1/4 cup out for the laces. Add the cocoa powder and mix until smooth.
Frost cookies using an offset spatula, and a small piping bag for the laces.