Preheat oven to 325 degrees F (165 degrees C). Prepare a 9-inch pie pan with cooking spray.
Whisk together the eggs, half-and-half, sugar, and vanilla in a bowl. Arrange the cubed pumpkin bread in the pie pan. Pour the egg mixture over the bread cubes.
Bake in preheated oven until a knife inserted into the center of the dish comes out clean, 30 to 35 minutes.
Combine the caramel sauce, heavy cream, and rum in a medium microwave-safe bowl. Heat on High in microwave for 45 seconds; whisk.
Drizzle sauce over bread pudding to serve.
1 1/2 cups half-and-half
1/4 cup white sugar
1 teaspoon vanilla extract
4 cups cubed pumpkin bread
2/3 cup butterscotch caramel ice cream topping
3 tablespoons heavy cream
3 tablespoons dark rum