Preheat an oven to 325 degrees F (165 degrees C). Combine flour, baking powder, baking soda, and salt in a bowl. Set aside. Grease and flour a fluted tube pan.
Beat 1 cup of butter, shortening, 2 cups of brown sugar, and 1 cup of white sugar with an electric mixer in a large bowl until light and fluffy. The mixture should be noticeably lighter in color. Add the bananas, followed by the eggs one at a time, allowing each egg to blend into the butter mixture before adding the next. Beat in 2 teaspoons of vanilla extract with the last egg. Pour in the flour mixture alternately with the milk, mixing until just incorporated. Fold in the pecans; mixing just enough to evenly combine. Pour the batter into prepared pan.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 1 hour and 20 minutes.
Meanwhile, prepare the glaze. Heat 1/2 cup of butter, 1/4 cup of brown sugar, 1/4 cup of white sugar, 1 teaspoon of vanilla extract, and heavy cream in a saucepan over medium heat. Stir continuously until the sugars have dissolved, then gently simmer for 1 minute. Pour over baked cake while still in the pan. Cool in the pan for 10 minutes before removing to cool completely on a wire rack.