Preheat the oven to 350° and lightly butter the sides of a 9-by-13-inch glass baking dish. In a medium bowl, whisk the flour with the baking powder and salt.
In a standing electric mixer fitted with the paddle, beat 2 sticks of the butter with the granulated sugar at medium speed until very fluffy, 5 minutes. Beat in the eggs one at a time, then add the vanilla. In 3 alternating batches, beat in the flour mixture and the milk. Transfer one-fourth of the batter to a medium bowl and stir in the melted chocolate. Beat the peanut butter into the remaining batter.
In a small bowl, blend the remaining stick of butter with the brown sugar and spread it in the baking dish. Arrange the bananas, cut side down, in the dish. Spoon dollops of the peanut butter batter into the dish and fill in the gaps with the chocolate batter. Gently swirl the batters together with a table knife.
Set a baking sheet on the bottom of the oven. Bake the cake in the center of the oven for 1 hour and 5 minutes, until a toothpick inserted into the center comes out clean. Let the cake cool on a rack for 10 minutes, then invert it onto a platter. Let the cake cool slightly, then cut into squares and serve.