Preheat your oven to 350 degrees, and grease a 9x13 inch pan.
In a medium bowl, whisk together the flour, salt, baking powder, and baking soda. Set aside.
In a large bowl, beat together the butter, peanut butter, sugar, and powdered sugar until fluffy. Beat in the egg, water, and vanilla extract.
Slowly add in the flour mixture, and stir until completely combined.
Spread the dough evenly in your prepared pan. Bake for 14-16 minutes, or until the cookies look just golden near the edges. Cool completely before frosting.
For the frosting:
In a medium bowl, beat together the butter, peanut butter, powdered sugar, vanilla extract, and salt until combined and thick.
Beat in the heavy cream until the mixture becomes fluffy.