As you are aware, in order to fly professionally in Europe you need two things, the right to live there and a pilot's license. In your case it appears that the license might be the more difficult to obtain. To get a job over there you need what is called a frozen-ATPL. This requires you to pass fourteen different written exams, each in a different subject. Most people take between six months to two years to complete them. Then you would need to do the flight training. There are schools in the US that are qualified to teach the European syllabus, an example is Naples Air Center in Florida. Once you have your license, I would say that it is easier to get a job as a low-time pilot over there than it is in the US. Keep in mind, however, that US major airlines have not hired in significant numbers for ten years. When those guys start to retire, the US will be the place to be. Here are a few websites that may help:
Naples Air Center - Full Time Groundschool
Professional Pilot Training (includes ground studies) - PPRuNe Forums
Dutch nationality law - Wikipedia, the free encyclopedia