Yes, you can transfer FAA to EASA and EASA To FAA. You do not have to do the whole course over. Your hours count. It would probably be a faster process to go from 0 to Commercial with the FAA and get a job and then start working on the EASA conversion. If you are motivated, you can go from 0 to commercial certificate as quickly as 6 months in the US, but your first job will probably not be flying a jet, but rather towing banners, flying skydivers, flying a PC-12 or caravan, or flight instructing.
If you do your training in Europe, your training will be closer to a year and a half but when you finish, you may have an opportunity to fly a passenger jet but you may end up paying for a type rating (which are much more expensive in Europe, approx €30,000). There are some legacy carriers in Europe that don’t make you pay for your type rating, but most new commercial pilots are going to the low cost carriers and paying for the type rating.