1. Preheat the oven to 375°F (190°C) and grease a 12×18-inch sheet pan.
2. In a mixing bowl, combine the dry ingredients: sugar and flour.
3. In a saucepan over medium heat, melt butter, peanut butter, and water, stirring until smooth.
4. Pour the hot peanut butter mixture over the dry ingredients and mix well.
5. Add eggs, buttermilk, and vanilla extract to the batter, stirring until fully combined.
6. Pour the batter into the prepared sheet pan and spread it evenly.
7. Bake in the preheated oven for about 20 minutes or until a toothpick inserted in the center comes out clean.
8. While the cake bakes, prepare the frosting by melting butter and peanut butter in a saucepan over medium heat.
9. Add milk or buttermilk and vanilla, stirring until fully combined.
10. Gradually whisk in powdered sugar until smooth and creamy.
11. Once the cake is done baking, remove it from the oven and let it sit for about 5 minutes.
12. Pour the warm frosting over the warm cake, spreading it evenly.
13. If desired, sprinkle chopped peanuts on top for extra crunch.
14. Allow the cake to cool slightly before serving warm or at room temperature.
15. Store leftovers at room temperature in an airtight container.
16. For longer storage, wrap the cake well and freeze for up to 3 months.