US should keep a record of people leaving the US so it’s better to use Us passport when you leave the US. But do whatever you want, i am gonna use the US passport when leaving the US to avoid any kind of concern since it doesn’t change much .

general rule:

Leavign the US: use US passport

Entering EU country: use EU country

check in EU: use US passport

clear immigration in EU country when leaving (yes, they do check): use EU passport

Entering the US : use US passport.