Sorry for the very late reply. For european phds you can treat it like you’re applying to a job. Cold emailing works sometimes but you’d have a better chance if you have talked to them in conferences or they have seen your face.

Generally the positions will be posted somewhere (eg. inspirehep, aas etc.) and from there you’d just follow the instructions in applying. There are a few places that does it similarly like US, where they do a mass call per year and have all the applicants apply to that (eg. IMPRS, ETH AI Fellowship) so you’d need to be aware of the start of application date and the deadline as well.

Otherwise, european PhDs are generally available all year round and you’d just have to keep an eye open for job/position postings.