Stir together rice, water, 1/4 cup white sugar, and 1 teaspoon salt in a saucepan. Bring to a boil over high heat. Reduce heat to medium-low, cover, and cook until tender. Remove from heat, let sit for 10 minutes, fluff, then spread onto a parchment-lined baking sheet. Refrigerate uncovered for 8-24 hours.

Stir yogurt, brown sugar, vanilla, cinnamon, and the remaining 1/2 teaspoon salt in a bowl. Add refrigerated rice and stir well to coat.

Heat 2 tablespoons butter in a 10-inch skillet over medium-high. Add rice mixture and flatten evenly into a disc. Cook for 10 minutes until edges are brown and crispy. Flip onto a plate, add 2 tablespoons butter to the skillet, and slide rice back in to crisp other side for 8 minutes. Slide rice onto a plate and set aside.

Sprinkle banana slices with 2 tablespoons sugar and use a kitchen torch to caramelize until golden brown.

Melt the remaining 1 tablespoon butter in the skillet. Add caramel topping and cook until bubbling. Remove from heat, add rum, and use a torch to carefully ignite. Swirl the skillet until flames disappear.