I have never heard of anyone being successful in trying to propagate puffballs, giant or otherwise. Most of the ones I've ever found were in or near pine trees, and always in areas where the ground had not been disturbed for many years. And usually in the autumn.
The spores don't mature unless the puffball dries out, at which point almost any touch will make it "puff", hence the name. Whatever is growing on your leftovers is probably not the same as the puffball, I'd throw it out.
By the way, when collecting smaller varieties, it is a good idea to slice them in half. if you see a homogeneous spongy white interior, its probably a puffball. If you see a cross section of a stem and cap DO NOT EAT IT! The deadly fly agaric and the death angel both have a shroud that is attached to the bottom of the stem when they first emerge. It tears away as the fruiting body matures, giving the mushroom the classic shape with the ring around the base of the stem.
There is a saying about mushroom hunters: There are old mycologists, and there are bold mycologists, but there are no old, bold mycologists.
The man who taught me about mushrooms never ate anything unless he could positively ID it as edible in three different published sources.
Happy hunting, but be careful.