Masters in Europe, PhD in US I have seen various questions regarding PhDs in the EU vs US, but none I've seen address this. The subject area is Maths/Physics. 

From what I've seen, in the EU you do a Masters (I'm soon to be finishing) and then PhD. In general this might take 4-6 years total for both. In the US you apply for a PhD after finishing undergrad studies and it takes 5-6 years total. There are classes in the first two years or so which make it roughly comparable overall to Europe.

My question: Is it possible to do a PhD in the US after completing a MSc in Europe, without have to go through the class component in the US? In other words, can one go straight to research?
No, you will have to take the classes.  There will always be exceptions but one reason for this is that most Math/Physics PhD programs have a qualifying exam.  If you are not familiar with this, it is an exam that everyone takes after the first (sometimes second) year when everyone has finished the required classes.  It is used to measure whether or not the student has the theoretical foundation to then move onto research.  Failing the qualifier may mean being forced out, though you can usually at least get a MS if this happens.