In a large mixing bowl or stand mixer, cream together sugars, butter and peanut butter. Add eggs and vanilla until mixed well.
Slowly add remaining dry ingredients and mix until a well blended dough is formed.
Evenly press dough into a well greased jelly roll pan (I used a 16x12x1) and bake for 15-18 minutes or until golden brown.
Remove from oven and immediately spread additional peanut butter over the top of HOT bars (be careful when spreading as to not break apart bar). Set aside and allow to cool for 45-60 minutes.
Frosting
In a large mixing bowl or stand mixer, combine all ingredients on low speed. When combined, increase speed and beat until smooth and creamy. Adjust milk for consistency as necessary.
Spread over top of cooled peanut butter bars. Slice and serve.
Recipe by Half-Scratched at https://www.halfscratched.com/lunch-lady-peanut-butter-bars/