How soon can I put in a package for a different MOS or apply for a broadening assignment?

There's no package for a different MOS. It all depends on what MOS you want to get into and your DEROS. DEROS drives everything when you're OCONUS. For broadening assignments you'll need to be an NCO for at least a year, bare minimum.

Head to the SORB office next to the Museum and Shoppette with a complete SF physical, go to Selection (SF/CA/PO), get Selected, good bye orders and hello reclass.

You're going to have to wait some time and hit a few milestones before you can request a broadening assignment. Specifically, you'll need to earn the rank of SGT first for recruiting. Check out this link for a list of requirements for recruiting.
https://recruiting.army.mil/recruiter/eligibility_requirements/
Once you meet all of those requirements, you just have to do your time in your overseas assignment then. However, it doesn't matter the time requirement for an overseas assignment, if you don't meet the basic qualifications for the assignment you're looking into.
I wish you the best of luck!