Preheat your oven to 350°F. Grease a 9x13 inch baking pan with nonstick cooking spray or butter.
In a medium-sized bowl, whisk together the flour, baking powder, and salt. Set aside.
In a large mixing bowl, cream together the butter and sugar until light and fluffy, about 2-3 minutes. Add in the eggs, one at a time, and beat until well combined. Gradually add in the flour mixture, alternating with the milk, until the batter is smooth. Stir in the vanilla extract.
Pour the batter into the prepared baking pan and smooth it out with a spatula.
In a separate bowl, make the peanut butter layer by creaming together the peanut butter, butter, powdered sugar, vanilla extract, and salt until smooth. Spread this mixture over the top of the cake batter in the pan, using a spatula to create an even layer.
Bake the cake for 25-30 minutes or until a toothpick inserted in the center comes out clean.
While the cake is baking, make the chocolate layer. In a double boiler or a heatproof bowl set over a pot of simmering water, melt together the chocolate chips and butter until smooth and fully melted.
When the cake is done, remove it from the oven and let it cool in the pan for 5-10 minutes. Then, pour the melted chocolate over the top of the peanut butter layer, using a spatula to spread it out evenly.