Whisk together flour, brown sugar, baking powder, baking soda, and salt in a bowl. Whisk together peanut butter, sour cream, butter, and boiling water in a large bowl (mixture will be very thick). Stir flour mixture into peanut butter mixture until fully combined, then stir in chocolate chips.
Generously coat a 6-quart oval slow cooker with cooking spray. Spread batter evenly in slow cooker.
Whisk together white sugar, cocoa powder, milk, and vanilla in a small bowl until smooth. Pour over batter.
Cover and cook on High until sides of cake begin to pull away from ceramic liner and appear solid (the center will still be soft), 1 1/2 to 2 hours.