Preheat oven to 350°F (180°C).

In a large bowl, whisk together mashed banana, melted ice cream, brown sugar, oil, sour cream, egg, egg yolk, granulated sugar, rum, and vanilla bean paste until well combined, stopping to scrape sides of bowl.

In a large bowl, whisk together 1½ cups (188 grams) flour, baking powder, salt, baking soda, and cinnamon. Gradually stir flour mixture into banana mixture just until some flour streaks remain.

In a small bowl, stir together finely chopped pecans and remaining ½ teaspoon (1.5 grams) flour. Fold pecan mixture into batter just until combined.

Spray a 4- to 5-cup Bundtlette pan with baking spray with flour. Divide batter among prepared wells. Forcefully tap pan on a kitchen towel-lined counter several times to settle batter and release as many air bubbles as possible.

Bake until a wooden pick inserted near center comes out clean, 12 to 18 minutes. (Cakes will bake with a slightly rounded top.) Let cool in pan for 20 minutes. Invert cakes onto a wire rack, and let cool completely.

Using a serrated knife, level bottom of cooled Bundtlettes. Place, cut side down, on a large rimmed serving platter. Using a spoon, drizzle Brown Sugar Rum Sauce over and into center of cakes as desired; garnish with scoops of ice cream, chopped pecans, and brûléed bananas, if desired. Drizzle with more Brown Sugar Rum Sauce; serve immediately with any extra sauce.