Comment by jmyeet
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.