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. Please see the notes above the recipe for tips on spreading the peanut butter. 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
Butter: I use salted butter in this recipe.
Brown sugar: Light brown sugar or dark brown sugar will work in this recipe.
Peanut butter: Use regular peanut butter (Skippy, Jif, etc.) in this recipe. The natural peanut butters without oil and sugar don't work well in this recipe.
Milk: Any milk, from skim milk to whole milk, will work in this recipe. Nondairy milk, like almond milk, will also work.
Powdered sugar: If your powdered sugar is lumpy, sift it before adding it to the frosting. This will help it mix in easily.