In a medium saucepan, melt butter over medium low heat. Continue cooking until butter has turned a golden-brown color. It should smell caramel-y. Remove from heat and stir in sugar and cocoa. Add vanilla. Stir in eggs on at a time, until well-incorporated. Stir in salt and flour until just combined.
Line an 8 x 8″ baking dish with parchment paper. Pour in brownie batter and smooth the surface. Bake brownies at 325 degrees for 20-25 minutes, or until brownies are set. (A toothpick inserted will still come out with a few crumbs) Allow brownies to completely cool before frosting.
Make frosting by beating butter, powdered sugar, peanut butter, and pinch of salt until light and fluffy. Spread over brownies. Slice brownies into 16 squares.