In cookies when you are baking without eggs you can substitute it with applesauce, banana, flax egg or chia egg. I have even used mayonnaise instead of eggs. It depends on the type of cookie you are making, these substitutes will make different textures. For these oatmeal cookies I used a flax egg as it creates a chewy texture. However, in my eggless brownie cookies I use a banana to make it more of a soft brownie texture. For the best results follow an eggless cookie recipe or a vegan cookie recipe.