Preheat the oven to 175°C conventional or 160°C fan. Grease and line a 23 × 13 cm loaf pan with baking parchment.
In a large bowl, whisk the melted butter, mashed banana, applesauce, and brown sugar together.
Add the eggs one at a time. Whisk in the vanilla extract.
In a separate bowl, combine the gluten-free flour, baking soda, cinnamon, ginger, nutmeg, and salt.
Add the dry ingredients to the wet ingredients. Stir gently until almost combined.
Fold in the shredded carrot and chopped walnuts. Do not overmix.
Let the batter rest for 5 minutes, then transfer it to the prepared loaf pan and smooth the top.
Bake for 55–65 minutes, or until a toothpick inserted into the centre comes out clean or with a few moist crumbs. Cover loosely with foil if the top browns too quickly.
Cool in the pan for 15–20 minutes. Lift the loaf out and cool completely on a wire rack.
Beat the softened cream cheese until smooth. Add the icing sugar, vanilla, and cinnamon.
Add the almond milk one teaspoon at a time until the frosting reaches a spreadable consistency.
Spread the frosting over the completely cooled loaf. Chill briefly for cleaner slices, then cut and serve.