Before any job issues, I would make sure you really want to move to Europe. Since you’re on visa, you may find it very difficult to return if you decide to leave - it can be quite difficult and competitive to apply for a USA job from abroad. Secondly, really make sure you would enjoy the place you’re moving to if things didn’t go well. To put it bluntly, your girlfriend today may not be your girlfriend tomorrow. Lastly, as you probably already know, you should expect a pretty steep cut to your salary.

Now, assuming you square away all those doubts, I recommend having a few prospects both domestically and in Europe before approaching your manager. In the event they react badly, you at least want to have another job option. It’s up to you if you want to have an offer or not, but i recommend it at least be interviewing when you speak to tie manager. Regardless of when, if you like your company, you’ll want to open by saying you want to stay and that you’re looking for a way do that while also being closer to people you live in Europe. Your chances are better if they already have employees there or your contributions have been exceptional. Good luck.