I am not sure if I should contact HR while on PTO just to give notice.
Contractually specified notice period is three weeks. If I wait till I come back, I will be giving them notice period of three weeks.
On the day you return from PTO, give your contractually obligated three weeks notice. Most companies will have a problem if you take significant amount of time off during the notice period. Though most don't define significant.
The notice period is to finish your current tasks, document what you do, and make yourself available to train somebody else. Though in most cases anybody you train will already be on the team, because it takes time to find a replacement, plus they might also have a notice period.
This really depends.
For example, in India, there is a rule in most of the organisations, that you cannot avail / use PTO while you're in a notice period. Hence, two of the things can happen here
where either of these have the same effect, just different procedure.
Your best bet it, submit the resignation already. Best case, the'll accept the same and consider the countdown from the date of submission, worst case, the'll start the countdown once you join, or ask you to re-submit when you resume work. You got nothing to lose by submitting it right now (while you're on PTO).