Flax/Chia egg would give a little bit of a dense texture to the carrot cake. I'd recommend 3 tablespoons of aquafaba ( the liquid left in a can of chickpeas) for each egg you're replacing (whip this up first before adding to your other ingredients). Another option if you can eat dairy products is to replace each egg with a quarter cup of yoghurt, whip a little first to give it a little volume (mixing it with the sugar & oil in a recipe with a whisk will help, before proceeding w/the rest of the recipe).
You could replace it with an equal amount of mashed banana, but this will come through as a taste in your cake of course, so be aware of this ! Other options are (equal amounts of - 1)another fruit puree 2)plain yoghurt 3)smooth nut butter. You can make your own apple puree easily by peeling & slicing an apple. Slice it thinly, add to stove with a small amount of water and cook until soft. Wait for it to cool & then mash with a fork/puree in a blender/with blending stick. If you want to add extra sugar to this do so while cooking on the stove, but apples are already sweet of course, so this isn't completely necessary.
A quarter cup of plain yoghurt, preferably with high fat content (to replace fat in egg). If you are vegan, you can replace each egg in a recipe with 3 tablespoons of aquafaba (the juice/liquid left after cooking dried chickpeas, or the liquid left in a can of cooked chickpeas). Whip it up slightly (w/ a whisk) first to give more volume to baked goods.
