Position racks in the center and bottom third of the oven and preheat to 350 degrees. Lightly butter two 9-inch round cake pans, sprinkle evenly with flour and tap out the excess.
Sift the flour, sugar, baking soda, cinnamon, and salt into a bowl. In another bowl, stir or whisk the bananas, pineapple, oil, eggs, and vanilla, until combined. DO NOT use the electric mixer.
Pour into the dry mixture and fold together with a large spatula just until smooth. DO NOT beat.
Fold in the pecans. Spread evenly in the pans.
Bake until the cakes spring back when pressed in the center, 30 to 35 minutes. Transfer the cakes to wire racks and cool for 10 minutes. Invert the cakes onto the racks. Turn right side up and cool completely.
Using an electric mixture on high speed, beat the cream cheese and butter in a large bowl until combined. On low speed, gradually beat in the powdered sugar, then vanilla, to make a smooth icing.
Place 1 cake layer, upside down, on a serving platter. Spread layer with about 2/3 cup of the icing. Top with the second layer, right side up. Spread the remaining icing over the top and sides of the cake. Store covered in the refrigerator overnight if not serving immediately. Let stand at room temperature before serving.