R.M.
Surely your 11 month old has had milk in some form during her life? My guess would be that it is the eggs, if this is the first time she has eaten eggs. But I am also sure she has had eggs, ...maybe not straight scrambled eggs...but eggs in a cake mix...eggs in a recipe of some sort.
I would talk to your pediatrician...do some research on egg allergies...and be on your guard. Also talk to your pediatrician about what you should have available to use if she has this type of reaction again. I don't know if she is old enough to be taking liquid benadryl or not but that is a standard antihistamine.
Good luck...you usually have to do a lot of detective work to figure allergies out.