In a large saucepan, bring 5 1/2 cups milk, sugar and salt to a boil over medium-high heat.
As soon as it comes to a boil, stir in the rice and reduce heat to low. (Note: You want it to simmer gently.)
Cook for 50 to 60 minutes, stirring occasionally. Note: You want to cook it until it thickens up like yogurt.
Stir in vanilla.Let cool and then refrigerate until chilled.
To make sauce, melt butter in a medium skillet over medium heat. Add brown sugar. rum, cream, and salt. Stir and cook for 2 minutes.
Add banana slices and pecans. Cook until bubbles form around the edges and bananas are coated well.
To serve, stir the remaining 1/2 cup of milk into the rice pudding.Dish into bowls and top with sauce.
Notes
The rice pudding is sweet and the sauce is very sweet. If you tend not to like really sweet desserts, try reducing the granulated sugar in the rice pudding to 1/3 cup.