Preheat oven to 375 degrees F. Grease a 9x13 baking pan with cooking spray and set aside.
In a medium bowl, whisk together flours, baking soda, and salt. Set aside.
In the bowl of a stand mixer, cream butter, peanut butter, and brown sugar together until smooth, about 3 minutes. Add in the egg and vanilla extract. Mix until combined.
With the mixer on low, slowly beat in the flour mixture. Mix until just combined. Stir in the oats.
Put dough into prepared pan. Press gently with fingers to even the dough out. Bake for 12-14 minutes or until bars are light golden brown and set around the edges. Cool completely.
While bars are cooling, make the chocolate frosting. In the bowl of a stand mixer, using the blade attachment, beat the butter and cocoa together until smooth, about 2 minutes. Alternately add powdered sugar and milk and beat until creamy and smooth. Mix in the vanilla extract.
Frost the cooled bars with chocolate frosting. Cut into squares and serve.