Preheat oven to 350 F. Grease a jelly roll (10x15-inch) baking pan. Set aside.
In a large bowl, beat butter and sugars together until well-combined, about 2-3 minutes.
Add eggs and vanilla, and mix until combined, scraping down the bowl as needed.
Mix in 3/4 cup peanut butter until well-combined, about 1-2 minutes.
Stir in flour, baking soda, and salt until almost incorporated.
Stir in oats until just incorporated.
Spread batter into prepared pan.
Bake 12-15 minutes, or until golden brown.
Let bars cool in pan for 10 minutes.
Beat remaining 3/4 cup peanut butter until whipped and fluffy.
Drop spoonfuls, and gently spread to cover bars.2 Set aside.
In a medium saucepan, add butter, milk, and cocoa powder.
Bring just to a boil over medium heat.
Remove from heat, and stir in vanilla and powdered sugar3 until combined.
Pour over bars.
Let cool in the pan completely before cutting and serving.
Notes
For this recipe, you'll want to use regular peanut butter with sugar. Natural peanut butter (just peanuts and salt) won't work well in these bars.
Spreading the peanut butter in one direction helps to keep the peanut butter from pulling up the bars.
If your powdered sugar is lumpy, sifting it before adding it to the icing will help it to stir in easily and will help keep the frosting from having lumps.
I prefer to chill the bars prior to serving, but they can be served cold or at room temperature. I also store them, covered, in the fridge.