Vida - I am not sure that I agree with your statement:
"For the return, use your EU passport to board and the US passport to enter the US and for any connected paperwork to fill out on board the aircraft that is for entry. "

I think that I should board with my US passport, as the airline has to ensure that I have either visa, green card or am a citizen to enter US.

Etoile - thank you, we just got the US passports, so all this dual thing is very new to me and yes, I am nervous not to make a mistake