Preheat the oven to 350F. Lightly grease a 17x12x1 sheet tray.
In the bowl of an electric mixer fitted with a paddle, cream the butter and the sugars together until light, fluffy, and creamy. Add the peanut butter, eggs, and vanilla and continue to mix on medium speed until fully combined and smooth.
Slowly add the flour, oats, baking powder, and salt and thoroughly mix until well combined.
Spread the dough onto the prepared sheet tray. Bake for 18-20 minutes or until golden brown. Remove from the oven and let cool to room temperature.
For the frosting, add the butter, cocoa, sugar, and vanilla to a bowl and beat until smooth. If the frosting is too thick, thin it out with the milk until you reach the desired consistency.
When cool, spread the frosting over the bars, cut into portions, and serve.