How do you say it correctly?
The publicly well-known project
The public well-known project
It depends what you’re trying to say. If you want to refer to a public project that’s well known, you need ‘the well-known public project’. If, on the other hand, the project in question is not necessarily a public one, but one which is nevertheless well known to the public, then it would be grammatical to say ‘the publicly well-known project’. Grammatical, but not very likely. What you’d be more likely to hear would be ‘the project well known to the public’.
The difference is that in the first, the adverb publicly modifies the adjective (well-known), not the noun. This is grammatically correct.
In the second, the adjective public modifies the noun (project), not the adjective. More common in this case would be to switch the adjectives (The well-known public project).
Due to the specific words chosen in this case, the meanings are almost the same. The difference becomes more apparent when you replace well-known with disapproved:
P.S. I've added hyphens to your well knowns; it's the correct way to do it.