Is it permissible to celebrate a child's growth and aging on a day other than their birthday?

We issue a fatwa prohibiting the celebration of birthdays because it is an unlawful 'Eid and because it involves imitating non-Muslims. If the celebration of a child is limited to a specific, recurring time, then it is considered an 'Eid and takes the ruling of a birthday celebration, because an 'Eid is what recurs from a general gathering in a habitual manner. However, if the celebration is incidental and not habitual, then it is not an 'Eid. Nevertheless, the matter of it being an imitation of disbelievers still needs to be considered, and the prohibited imitation is that which is specific to them. It is more prudent to avoid such celebrations out of piety.