Egg sensitivity is not unusual in young children, and in fact her sensitivity to eggs by 2 years of age, was one of the first warning signs that food allergies were one of the important things to consider when making birthday cakes for her or other family members.
She's okay with egg inclusion in recipes now, but is definitely allergic to nuts of all kinds.
However her younger brother, almost 2 is sensitive to eggs. Luckily for me, during the last 8 years I've learned how to adapt some recipes and even found a few that were egg free with regard to cakes. Google recipes for Chocolate Pudding Cake as an example... great as if you are a grandma, probably your pantry will contain most or all of the ingredients. This one is a great reward for snow shovelers too, and takes not all that much time to prepare and bake.
Recently, I heard a reference to AquaFaba spelling may be incorrect, but google can set that straight too. http://www.glueandglitter.com/main/2015/04/14/aquafaba-magical-egg-replacer-for-vegan-meringue-recipes-and-beyond/ Meringue? Are you kidding me?
I'm making "you gotta be kidding me cake" for her birthday, normally easy peasy to do, but without the eggs? So I looked into aquafaba a bit via google, and found out (at least in my head, that the drained fluid from 1 can of garbanzo beans = approx 1 egg.
Since 3 eggs were needed, I decided to make a crunchy garbanzo bean snack for all of us to enjoy too.(not necessarily with the cake) I added the crunchy beans to an "Earth Day Salad" Livvie's naming of a green salad with carrots, celery, etc. and also just had some available for taste testing.
When I drained 3 cans of generic garbanzo beans I got about 1 quart of fluid. Having baked numerous birthday cakes in my 69 years, that looked like WAY TOO MUCH liquid! Turned out the straight substitution of aquafaba for eggs, was too much as this particular recipe also has a can of cherry pie filling.... oh, yeah, baby!
http://www.thekitchn.com/medium-large-jumbo-how-egg-sizes-actually-measure-up-ingredient-intelligence-200891 Tried to link it, but somehow or other when I copied the URL something went missing. The information below was copied directly from that site, so I am including the link above for your further information(great photos and narration!) but also to *footnote or give credit to the person who conducted the actual investigation.
So, to summarize I needed egg substitute for 3 large eggs and egg volume in liquid measure for eggs- "Large: 3 1/2 tablespoons (1.75 liquid ounces)" so times 3 eggs would be 10.5 tablespoons according to my head math, verified by a calculator. Will try to remember to add a photograph of cake to post....