I was concerned about this myself, but I was lucky to be able to schedule both doses in the EU way ahead of my trip. If your national vaccination system allows rescheduling appointments, my suggestion is to keep checking regularly for any earlier dates (in some countries this is allowed and earlier timeslots are released as people cancel/reschedule appointments).

Another obvious option is to delay it and get both doses in either the US or EU, but getting on a TATL flight unvaccinated is semi-suicidal, so that's a no go, way better to have at least one dose.