Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch cake pan.

Melt 1 tablespoon of butter in a small skillet over medium heat. Stir in apple, raisins, 1 tablespoon rum, and a pinch of cinnamon. Cook until apples are soft, about 10 minutes. Toss apples with bread cubes, and spoon into prepared pan.

Combine brown sugar, white sugar, 3/4 cup milk, evaporated milk, 1 tablespoon cinnamon, and 1/4 cup butter in a small saucepan over medium heat. Cook, stirring, until butter is melted, about 5 minutes.

Whisk eggs and 1/2 teaspoon vanilla extract in a small bowl. Whisking constantly, pour hot milk mixture into beaten eggs. Pour custard over bread cubes and apples.

Cover pan with aluminum foil; bake in preheated oven for 40 minutes. Remove foil and bake until pudding is puffed and lightly browned, about 10 additional minutes.

While pudding bakes, prepare rum sauce. Whisk together 3 tablespoons milk and cornstarch in a small bowl until smooth. Melt butter over medium heat in a small saucepan. Stir in 1 tablespoon rum, 1/2 cup milk, and 1/4 cup white sugar; bring to a boil. Slowly add cornstarch mixture, stirring until sauce thickens. Remove from heat and stir in 1 tablespoon vanilla.

Pour rum sauce over warm pudding. Let rest for 5 minutes before serving. The sauce will continue to thicken and form a beautiful glaze.