I think EU should make it easier for US citizens who get any job here to move over. Americans in general are super hard working. Europe needs hard working people, while we can provide them with free universal healthcare, free university education, and way more upward mobility than they could ever dream of.
The deal could be so that you can move to Europe for any job, and if you work for 5 years then you are eligible for permanent residence. If you loose your work, and become unemployed for more than 6 months you have to move back to US.
Yes, there would be risks, but at the same time, it could also be a win win - hard working folk from USA would get a chance to raise their kids in a society that would allow them to put their kids through college without selling a kidney.