Preheat oven to 350°F (180°C).

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

In a large bowl, whisk together 3 cups (375 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 1 teaspoon (3 grams) flour. Fold pecan mixture into batter just until combined.

Spray a 10-cup Bundt pan with baking spray with flour. Pour batter into prepared pan. Forcefully tap pan on a kitchen towel-lined counter several times to settle batter and release as many air bubbles as possible.

Bake for 35 minutes. Rotate pan and loosely cover with foil; bake until a wooden pick inserted near center comes out clean and an instant-read thermometer registers at least 200°F (93°C), 25 to 30 minutes more. (Cake will bake with a slightly rounded top.) Let cool in pan for 20 minutes. Invert cake onto a wire rack, and let cool completely.

Using a serrated knife, level bottom of cooled cake. Place cake, cut side down, on a 12-inch rimmed serving platter. Using a spoon, drizzle Brown Sugar Rum Sauce into and over grooves of cake as desired. Pour ½ cup (140 grams) Brown Sugar Rum Sauce into center of cake; 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.