Doing part of my PhD remotely in Europe while working full time?
You would need to check that with both the university and with any supervisor/advisor you would have. I'm pretty sure that such relationships do occur, even if not common. But being less the norm, you need to work out the details. 

In some ways the supervisor might be the bigger block than the university. It might take a trip to Europe to finalize any such arrangements. But you first need to determine if they would even consider such a situation. The answers will vary.