So far we have been doing great eating all of the whites, and probably half of the yolks, and feeding the remaining (cooked) yolks (and shells) to our finches and back to the chickens. It is working out well. I plan to make spaghetti carbonara sometime though! It looks good.
I don't think food banks and homeless shelters here would take eggs from us.
There is always a race for the leftovers on the Carbonara around here,where leftovers are usually not a fave amongst the children, there are a few but not many. Did you ask any church or local food pantries? The one in my town will take anything you bring and they always say thank you to all the extra produce. Usually sell all my eggs but they have taken some when I was back logged. Have you tried selling any?