Preheat oven to 400 degrees. Grease a rimmed half sheet pan.
Add the sugars, flour, baking soda, and salt to a mixing bowl and whisk to combine.
Add 1 cup of milk, peanut butter, and butter to a small sauce pan and bring to a boil, stirring often.
Remove from heat and let set for 5 minutes to cool. Add to the dry ingredients.
Add the remaining 1/2 cup of milk, vanilla, and eggs to the bowl and use an electric mixer to beat together on low speed until the batter is smooth and well combined.
Spread cake batter into prepared baking sheet and bake for 15 minutes. Cool completely, about an hour.
To make the frosting:
Add the sugar, butter, and milk to a small sauce pan and stir well. Bring to a boil and boil for exactly 60 seconds.
Remove from the heat and stir in the peanut butter and chocolate chips until smooth and melted.
Pour warm frosting over the cooled cake and spread with a spatula. Frosting will harden as it cools.
Cut and serve once frosting has cooled and set.
Notes
Be sure to use a processed peanut butter, like Jif or Skippy, and not the natural kind that separates and has to be stored in the fridge.Be sure to cool the peanut butter mixture a bit before mixing with the other ingredients so that you don't accidentally scramble the eggs with the hot mixture.The frosting will seem a bit runny, but it will firm up as it cools. For a thicker layer of frosting, double the frosting recipe.