Preheat the oven to 350°F (175°C). Grease and flour two 9-inch round cake pans. In a large mixing bowl, combine flour, sugar, brown sugar, baking soda, cinnamon, nutmeg, and salt. Use a whisk to evenly distribute the dry ingredients.
In a separate bowl, beat the eggs with a hand or electric mixer. Add vegetable oil and vanilla extract, mixing until smooth. Stir in mashed bananas and crushed pineapple until fully incorporated.
Gradually pour the wet mixture into the dry ingredients, mixing gently until just combined. Fold in chopped nuts with a spatula for even distribution.
Divide the batter evenly between the prepared cake pans. Place in the oven and bake for 40-45 minutes, until the tops are golden and a toothpick inserted into the center comes out clean. The edges should slightly pull away from the sides of the pans.
Remove the cakes from the oven and let cool in the pans for 10 minutes. Transfer to a wire rack and cool completely before frosting.
Once cooled, spread cream cheese frosting between the layers and over the top and sides of the cake. Slice and serve.
