Its been really mild in NY and I have pips on all my outdoor peonies, but today I bought a new cultivar, bare root, and its got three eyes, but they are already turning green and starting to open.
I'm wondering if I should plant it outside, or pot it for now. I usually plant pretty deep, but since these growths have already started to unfurl, I'm reluctant to bury them more and an inch or so, and I don't think that's going to be deep enough long term. Im temped to find a 10" pot and plant it shallow for now to maximize the chance it will bloom this year.

What are your thoughts? Plant in ground or pot? if pot, keep inside or out?
I was thinking of potting it to a dept with only maybe an inch over the tips and putting the pot outside so it doesn't rush (I don't have good inside light, so I don't want it to think its time to rush) but there will be many more frosts before real spring. Then maybe late April, when its 6-7" tall, burring the root (and obviously a good portion of the stems to a normal depth (the tuber ~6" down) outside.

THoughts?

I'd put it in the ground oriented sitting just a tiny bit further back than your pic shows so the growing tips are pointed up and the main feeder root is laying flat horizontally. The top of the crown where the new growth is emerging should be around 2" below ground for your zone but you can actually plant it a bit shallower and plan to mound soil to get it to the proper depth in the fall or just add a bit more compost for the winter. I wouldn't leave it in a pot outside in zone 5 or below or anywhere expected to get hard freezes still to come this spring. If you pot it up put it in a protected spot in the garage or a shed, some where that the pots soil won't freeze.
I'm in zone 4b and that's how I'd treat it, if there's someone with experience in your specific area I'd listen to them!