Actual US to EU transplant here, although not one who needed a visa. God bless Irish ancestry.
The most efficient way would be to simply let US citizens post a bounty for e.g. a successful job offer being extended their way from the European Union. When I think about the requisite legwork involved from my end, I come up with a rough goal of about $25,000 per job offer floated, not contingent on the person on the other end actually accepting the offer. (This is not a serious offer, please do not email me asking me to help you find a job.)
The reason the number is so high is because, frankly, it takes a lot of work to unearth opportunities like this which aren't already on the clearnet and inundated with applicants already. The dropping of the language barrier alone raises the competitive pressure on any job offer in Europe by an order of magnitude, in both salary and applicant quality.
Uniquely, US citizens get paid much more in almost every category than their European counterparts, so job offerers also have to factor in an exit risk they don't normally have to with applicants from e.g. Laos: the very real possibility that the starry-eyed rich kid shows up, spends 3 months living the way the Romans do, and then leaves because Rome kind of sucks compared to having a 401(k).
$25k doesn't seem that expensive when compared to, say, the opportunity cost of first learning e.g. passable professional German, which by itself takes about 1-2000 hours of one's free time. Someone getting paid even $50/hour = $100,000/year in the US would quickly eat way more cost going about getting here in such a roundabout way, and then they still have to find the visa grantor.
Again: This is not a serious offer. Please do not email me asking me to help you find a job. This was a thought experiment for curiosity's sake only.