When should I plant my sprouting potatoes...

I have potatoes which have been sprouting for about 6 weeks or so. They have been growing in about 50-55F temperatures. In the last 3 weeks, they have started to quickly deplete the parent potatoes. A local planting chart lists this month as a month one can plant potatoes. However, we could always get late frosts so I guess I'm wondering how frost tolerant potato plants might be. Thoughts? Below are my shoots...

They are nightshades so I hate to be the person to tell you wait until 1st frost free date in your area and try planting out.

You can go well before your last frost date. I plant my early potatoes out 2 months before last frost. they have a bit of protection from either fleece/frost netting or a low tunnel and that is all they need. The rest of my potatoes are planted around 1 month before last frost, it takes potatoes a few weeks to get out of the soil and even if there is a frost that takes out the tops they will regrow without an issue.

So you have 2 + weeks from the day you plant them until they break the surface and even if they get frost burnt they will happily come back, I will say that your tubers are drying out rather than anything else so a good soak for a few hours would be a good idea.

Agreed, if you plant them and they freeze off, they will come back.

But I see no reason to panic. Some of my potatoes have much larger sprouts. I will just break them off and new ones will form in time for planting in two months' time.

Soak them and then plant them whole or can I split those potatoes to make more plants?

You could cut these in 6 or 8 pieces if you wanted too.   Quarters should be great. let them heal over for a day or so after cutting them.

I live a bit north of you, and I think I have usually made the mistake of waiting too long to plant potatoes.  The ones that did the best last year were a couple that accidentally overwintered and got a jump on the rest that I planted in early April.  Plus, the overwintered volunteers did not get any pampering-- no trenching, no extra layers of compost.  

I don't think all potatoes reliably overwinter in our climate, but the fact that some can shows that it's okay to get them out early.