As a US citizen, you are supposed to show your US passport when exiting and entering the US.
EU countries don't require a visa for US citizens to visit, so showing your US passport would be sufficient for both showing you are in the US legally and that you have authorization to visit the destination country.
Then when you arrive in the EU, show either your US passport or EU passport depending on the rules of the particular country. If you are a citizen of the specific EU country you arrive in, their rule probably will be that you must show their passport to enter, not a US or any other passport.
But if you arrive in another EU country of which you are not a citizen, you can show either passport -- unless the EU in general has a rule that if you are an EU citizen you must use an EU passport to enter any EU country when coming from outside the EU.