Preheat oven to 400°F. Grease a large cookie sheet or line with parchment paper.
Mix together dry ingredients: flour, sugar, baking soda, and salt. Set aside.
Whisk together ½ cup milk, vanilla extract, eggs. Set aside.
In a saucepan, melt 1 cup (2 sticks) of butter and 1 cup peanut butter, bring to a boil. Remove from heat and slowly add flour mixture, followed by milk and egg mixture. Mix until well combined.
Pour batter onto cookie sheet, spreading batter evenly to edges. Bake for 20 minutes or until toothpick inserted in center comes out clean. Cool completely before frosting.
In a saucepan, bring 1 stick butter and ½ cup peanut butter to a boil. Add 2 tablespoons milk and slowly mix in powdered sugar until well combined. Pour frosting over cake, spreading evenly to edges. Allow frosting to cool and set before serving.