Hmm.
You could probably get away with substituting finely shopped feijoas for either banana or carrot in their respective cakes.
They are way sweeter than bananas though, so you'd want to cut back on the sugar there.
Here's a recipe I found:
Feijoa Cake Recipe Ingredients
1/2 cup milk
2 eggs (beaten)
1 cup white sugar
1 cup mashed feijoas
1 tsp vanilla essence
75 grams butter, softened
2 cups flour
1 tsp baking powder
1 tsp baking soda
Method
1. Cream the butter and sugar, blend in the feijoa.
2. Heat the milk, add the baking soda. Add to the mixture above, along with the eggs and vanilla essence.
3. Sift the flour with the baking powder, mix into the wet ingredients.
4. Pour into a greased and baking paper-lined 20cm cake tin and bake at 180°C for 40 minutes.
Ice with cream cheese icing if wished.
Cream Cheese Icing
200g Cream Cheese
200g Icing Sugar
1 T butter
some lemon juice
Beat Cream Cheese until soft, add icing sugar, lemon and butter. Blend till smooth.
