Preheat the oven to 350 degrees F. Line a 10-by-15-inch jelly roll pan with parchment paper and spray with cooking spray.
Whisk together the flour, baking powder, baking soda and salt in a bowl. In a measuring cup, combine the buttermilk, oil and vanilla.
Beat the eggs and sugar together in a large bowl with an electric mixer until very pale and thick, about 5 minutes. Add the flour mixture in three batches, alternating with the buttermilk mixture, beginning and ending with the flour.
Pour the batter into the prepared pan and smooth evenly with an offset spatula. Tap the pan on the counter lightly to release any large air bubbles, then bake until golden brown and a wooden pick inserted into the center comes out clean, 20 to 25 minutes. Dollop the peanut butter on top and evenly smooth it over the cake as it melts. Transfer the cake to a cooling rack and cool completely.
Put the chocolate chips in a microwave-safe bowl and microwave, stirring every 30 seconds, until melted. Pour the chocolate over the peanut butter and spread it evenly with the offset spatula. Refrigerate until just set, about 20 minutes.
Cut the cake into squares and refrigerate until ready to serve.