Preheat your oven to 350 degrees. Butter two 9-inch round cake pans (at least 2 inches tall) and line the bottoms with parchment paper. Butter the parchment paper and lightly dust the pans with flour, shaking out the excess.
In a small mixing bowl, stir together finely chopped pineapple with 1/4 cup all-purpose flour. Set this mixture aside.
In a medium mixing bowl, whisk together the remaining 2 1/4 cups of flour, baking soda, cinnamon, nutmeg, and salt.
In a separate large mixing bowl, using an electric hand mixer set to medium-high speed, whip together granulated sugar, brown sugar, and eggs until the mixture becomes pale and fluffy, which should take about 4 minutes.
Slowly blend in the vegetable oil, and then mix in the vanilla extract.
Add half of the flour mixture, and using a rubber spatula, fold it in just until combined. Add the mashed bananas and fold to combine, then add the remaining flour mixture and fold until combined. Finally, fold in the pineapple mixture and chopped pecans.
Divide the batter evenly between the prepared cake pans.
Bake in the preheated oven until a toothpick inserted into the center of the cake comes out clean, which should take about 40 minutes.
Remove the cakes from the oven and allow them to cool in the pans for about 10 mins. Afterward, run a knife around the edge to loosen them and invert the cakes onto wire racks to cool completely.
Frost the cakes with cream cheese frosting and gently press coconut onto the sides. If desired, decorate the top with a ring of whole or chopped pecans near the outside edge.
For the cream cheese frosting:
In a mixing bowl using an electric hand mixer fitted with a paddle attachment, whip together the cream cheese and butter until smooth and fluffy.
Add powdered sugar and vanilla and blend again until the mixture becomes whiter and the frosting is fluffy.
To firm up the frosting a bit, freeze it in 3-minute increments, mixing between increments until it reaches a slightly stiffer consistency.