My lo was not ready for egg whites on her first birthday. My mother found out you can substitute two egg yolks for one egg in a recipe. Then she made her carrot cake cupcakes. They had cream cheese frosting.
I am glad she chose cupcakes because they were easy for her to hold. For my sister and I, my mother would make a full-sized cake, take us outside in a highchair and let us tear into it. She wanted to do this again but I didn't really like the idea of my lo face-first in a cake. Cupcakes were nicer and everybody got to share them.
We did put candles on them. We did one cupcakes for each month so there were 12. She didn't blow them out, of course, but she had fun looking at them. We saved the candle that is shaped like a 1.
There are allergies in my family so I wanted to be careful. I would be more concerned about a possible reaction to chocolate than to eggs, however. Whatever you do, I'd avoid chocolate and nuts. Eggs--meh--probably be okay.