I've been growing paw paw for several years. They are just about everywhere on my property. I give them the same winter treatment as the chestnuts I grow but they take a lot longer to germinate in spring. First year, I get 12" growth, then they jump to 4-5' the second (potted up to 3g root pouches after a year in 1g).
Once I collect my seeds, I put damp long leaf sphagnum in a clear plastic shoebox about 3" deep. Place the seeds on this, then put another 1-2" layer of damp sphagnum on top. Add the lid and put in the fridge for the winter. I move the box to a sunny window in February and wait for germination. Unlike chestnuts (which I pot at first sign of radical), I let paw paw sprout and grow in the box. They often take until May/June to show much growth, but patience gets a high % to sprout. Once they get decent size, I'll gently untangle them and pot directly to 1 gal.