Comment by proberts
Broadly, an E-3 visa holder can pursue a green card and we sponsor a lot but there are potential issues, most notably, when seeking to renew an E-3 visa while in the green card process. Alternatives to the PERM (labor market test) based green process are the national interest waiver and extraordinary ability paths, which are not tied to a specific job/employer but require a high level of achievement.
Not a lawyer but I went through exactly this route back in the Obama administration so it may have changed. But even then there were questions about whether you could file an I-485 while on what was technically a nonimmigrant intent temporary visa (this includes E3 and TN). There were a lot of people who said it wasn't possible. Those people were wrong.
Here's how I approached it: when my PERM was approved, I immediately went and renewed my E#. This required a trip out of the country. I went back to Australia. My employer was amenable. This was to avoid being out of status or needing a renewal when my I-485 was pending.
One thing to note is that the application for the E-3 doesn't ask you "do you intend to immigrate?". It asks "Have you filed an I-485?" and at that point I could truthfully answer no because I hadn't.
I arrived back in the US and almost immediately filed the I-140 and I-485 concurrently.and applied for EAD and Advanced Parole. Those arrived within a few months.
The only wart was USCIS asked me to file an extra form giving up my diplomatic rights because other E visas are for diplomats. My lawyer said this really wasn't necessary but the best path was simply to give it to them. I think I got my green card after 8 months.