Preheat oven to 350°F. Line or grease a 9x13 baking pan.
Beat butter, both sugars on medium speed 2 to 3 minutes until pale and fluffy. Add peanut butter and beat 1 minute. Add eggs one at a time, then vanilla.
Whisk flour, oats, baking soda, and salt. Add to wet mixture and mix on low until just combined.
Press dough evenly into the pan. Bake 18 to 22 minutes until edges are set and center looks barely underdone. Cool completely.
Beat frosting butter and peanut butter until smooth. Add cocoa, vanilla, then alternate powdered sugar and milk until thick and spreadable.
Spread frosting over cooled bars in one direction. Chill 20 minutes, then slice into rectangles with a warm clean knife.