Would I need to also do the JOC course and MCC over there, or is that yet another EASA add on?
That is an EASA thing. In the states it is more of a gradual building up to it. Guys in Europe can end up in the right seat of a 737/320 fresh out of flight school so it does make sense. Don't read that as likely to end up there, most do not.
In the states your first job would more than likely be in a single engine piston requiring no additional training other than with the operator. If its over 200hp you need to do a short, couple of hours, training with ground school to be qualified to fly a high performance aircraft.
If it is pressurized a similar course.
You would, after those two short courses be qualified to fly almost anything up to 12,500lbs with piston or turboprop engines.
After that it is a type rating for a specific type.
Most in the US start either instructing or skydive flying or similar light piston job for a year or two. Then cargo or pax (right seat or left depending on experience) in a bigger piston twin or turboprop. Then right seat in bizjet or regional. Then a larger airline.
It is a path you can expect to follow if you are driven and have the aptitude.
In Europe its a lucky few that get into an airline straight away, a few other lucky ones get a job in a prop plane. Most do not end up working, try to stay current for a few years then give up. This is the reality of the situation in the EU, you may be lucky so its a gamble you have to decide weather or not to take.