Melt butter in a large skillet over medium-low heat. Add brown sugar and rum, if desired; stir until smooth. Remove from heat and stir in bananas.

Scatter half of bread in a 9x13 inch baking dish. Spoon caramel-banana mixture evenly over bread. Top with remaining bread. In a large bowl, whisk eggs until blended. Whisk in half and half, vanilla and salt. Pour over bread mixture, pressing down on bread to make sure it's all coated. Cover with foil and refrigerate at least 4 hours or overnight.

Preheat oven to 350 degrees. Let baking dish stand at room temperature while the oven preheats. Remove foil and bake until puffed and golden and a thin knife inserted in center comes out clean, about 40 minutes. Let cool for 5 minutes before serving.