Hi and welcome to the forum.

In your situation, the "best" way is probably a combination of approaches. If you can find employers in your field who post their available jobs through an agency, then by all means, contact the agencies. But it's also worthwhile to apply directly to employers for jobs - using online job posting sites or the online versions of newspapers and professional publications with job listings.

Speaking from experience, though, (going the other way - to Europe from North America) I can tell you that most employers are going to be hesitant to hire anyone sight-unseen from overseas - especially since they will have to go through a certain amount of hassle and expense to sponsor your visa application.

If possible, you should find a way to make yourself available for interviews once you get any sign of interest in response to your job query letters. I always included a line in my cover letter to the effect that I was planning a trip in their area in the near future, and would like to meet in person to discuss the position in greater detail. It's not cheap - but the first reply I had asking about when I could come in for an interview, I booked a flight over, and then recontacted some of the other employers, giving the specifics of my travel plans. I had a very busy two weeks of "vacation" and wound up with two competing job offers.
Cheers,
Bev