Is selling broken eggs from a nearby farm without the owner's knowledge and benefiting from the money generated thereby permissible or forbidden, and how can one atone for that and purify the money?

If the farm owner ordered the disposal of the broken eggs, then your friend is not at fault for what he did. However, if he did not order that, then your friend must repent and return the value of the broken eggs to their owner, in accordance with the verse: "Indeed, Allah commands you to render trusts to their rightful owners" and the hadith: "The hand is responsible for what it takes until it returns it."

If he sold the broken eggs mixed with the sound ones, deceiving the buyers, then he must compensate them for the deficiency. If that is not possible, then he must spend the money for the benefit of the Muslims. As for the profit resulting from investing unlawful money, there is a difference of opinion among scholars regarding it.