Preheat oven to 350 degrees. Butter the bottom of an 8 inch square baking pan with 1 tablespoon of butter and sprinkle with 2 tablespoons of sugar. Place the bread pieces in the bottom of the pan. Dot with the remaining butter and sprinkle with 1/4 cup sugar. Slice the bananas lengthwise into halves. Place on top of the bread and sprinkle with the rum. Spread the apricot preserves evenly over the bread and banana mixture. Ladle condensed milk over the bread.
In a medium sized bowl, whisk together the egg yolks and whole eggs. Heat the milk and remaining sugar until the mixture begins to simmer. Pour the milk slowly into the eggs, stirring constantly until well blended. Pour over the bread and banana mixture. Sprinkle with cinnamon and nutmeg. Place a heavy platter or cutting board over the pan to weigh down the mixture. Allow to sit for 15 minutes.
Bake in the preheated oven for 25-30 minutes or until an inserted knife comes out clean.
From "Tropic Cooking/by Joyce LaFray Young" / typed by Diane Newbury Recipe is credited to Henry Towle a chef in Coral Gables, FL.