A. Because of the risk of food allergies from eating egg whites, most experts recommend not giving whole eggs to children under twelve months of age. However, it is usually considered to be okay to just give them just egg yolks, once they are 7-10 months old.
You don't have to be in a big rush to give egg yolks though. According to the AAP, 'though in the past egg yolks were recommended as a good source of iron, the iron they contain is not easily absorbed. Iron-fortified cereals and meat are the best sources of iron for your infant.'
On the other hand, in addition to iron, egg yolks can be a good source of DHA, an important component of breast milk which is now being added to many infant formulas and baby food to try and promote a baby's growth and brain development.
So yes, you can feed your 11 month old scrambled eggs, as long as you only use egg yolks (and some formula, cheese, and butter of course).
Because of the risk of food poisoning from Salmonella, be sure to cook the egg yolks thoroughly though.