I recently had my first pawpaw fruit.  Apparently they grow locally here (NC mountains), but I had never heard of one.  I am by no means an authority, but this is what we did with it:

We just ate it raw.  It was very ripe (several large browning spots on the skin) and it was heavenly.  Very sweet, kind of melt-in-your-mouth mango-y.  I have no idea if you can cook with them, but they were so wonderful raw, I don't think I'd get that far.  Plus, it was so soft raw, I'd think it would disintegrate if you cooked it.  It also strikes me as a PITA to seed, if you were to cut it up to cook it. We just peeled it, stuck chunks of the fruit in our mouths and the flesh dissolved leaving the seeds.  I kept the seeds to see if I could sprout them.

The seeds are really big (think peanut M&M's).  The PITA part is that the flesh would kind of disintegrate if you cut them out.  Like an over-ripe mango where the flesh just sort of shoots through your fingers when you try to cut it from the seed and the seeds aren't located symmetrically or centrally through out the fruit. I cut off chunks of the flesh and spat out the seeds .  I don't think it is any relation to a persimmon.

In case you missed this site http://www.petersonpawpaws.com/Facts.php It says that because the flavors are so volatile, recipes should involve little or no cooking.

Once the jam is cooked and ready to bottle, picking the seeds out wouldn't be that big a problem, unless you simply don't want to do it. You could even put the jam through a china cap to strain out the seeds.

I never have seen higo chumbo (prickly pear) jam but that doesn't mean there isn't one. There's a place in my town that makes their own PP icecream. I've never tasted the pears because the guys who sell them on our streets aren't big on hygiene, and they sell them ready-peeled in plastic bags.

I think we're all talking about different things.  This is my understanding of the paw paw:
http://www.veggiegardeningtips.com/stalking-the-exotic-and-wild-paw-paw-tree/

At the end of the link above a poster commented about making cookies and freezing the juice with OJ.  interesting.

"Tired of making cookies. Now, I am peeling them, removing the seeds, putting the flesh in the blender with about a half cup of frozen orange juice concentrate and freezing the puree in pint jars. I’ll add it to oatmeal, breads, pancakes, and other such this winter. According to the Kentucky web site, it is very rich in vitamins and minerals, so don’t want to waste any."

OH! Pardon me all to pieces, nutdragon. That looks like a smooth-coated chirimoya, and not at all what I was thinking of. If that is a pawpaw, then the best way to process them would be to cut them open, dig out the flesh and pick out the seeds first and then proceed to do whatever, as the "cookie" poster says. Once de-seeded, I should think you could make jam...but you might have to add another fruit to get enough pectin for it to "jell." I don't know...how much pectin's in a paw-paw. (Which sounds like a tongue-twister along the lines of "How much wood would a wood-chuck chuck..." Sorry. I think I have a fever.)