It should be noted that the reason for the difference between the US and European style programs is primarily philosophical, and has to do with the way master's degrees are considered.
In Europe (or at least those countries part of the Bologna treaty), master's degrees "follow on" from the bachelor's degree program, and usually programs are designed so that students who complete the bachelor's degree will continue on to the master's at the same institution before beginning a doctoral program.
By contrast, in the United States, the master's degree is very weakly coupled to the bachelor's program. Instead, in many places, it's viewed either as a separate degree in its own right, or as a stepping stone to doing a doctoral degree. Moreover, because the bachelor's degree program is only four years, while the European bachelor's-master's system includes five years of coursework, there is a bit of a discrepancy between the coursework a bachelor's degree holder in the US would have, versus that of a master's holder in Europe. Consequently, most schools tend to require roughly a year of coursework for students entering a doctoral program, as it acts as completing the master's program at a European university.
Also, because graduate admissions are almost always organized at the departmental level in the US, rather than the research group level, there is usually a "qualification" procedure which must be completed at American universities that aren't found in European universities. The coursework phase of the doctoral degree often figures into the material tested in the qualifying exam, and therefore schools often are reluctant to waive these coursework requirements.
However, reluctance is not necessarily the same as refusal. If you have questions about how things work, and whether a particular department would be willing to waive some of the requirements for you, you should contact them. If you can demonstrate that you have most of the work already in place, they may be able to let you skip some of the classes, or at least replace them with other electives (which may be of benefit—you shouldn't assume you're done with classes and learning just because you have a master's or even a doctoral degree!).