If you have someone with egg allergies that you cook for often, then you would have heard about doing this. You can definitely use banana, applesauce, pureed apricot or even soft tofu to replace an egg. Use 1 very ripe banana smashed or 1/4 cup of applesauce for 1 egg. Banana has a stronger flavor, and you may taste it so be sure the flavor of your baked goods go well with it. I would use applesauce over bananas. Also add an extra 1/2 tsp baking powder to the recipe to help with rising.