Slughorn explains, when everyone is assembled, that alchemy with an egg is actually where phoenixes are speculated to originate; but no one has the recipe any more and there haven't been new phoenixes for centuries. They will not be making living things today. But the eggshell is a good base from which to appear lots of possible objects; alchemy doesn't like to conjure things into existence in plain sight but is quite willing that new things appear inside eggshells, cocoons, geodes, and similar. Their eggs have been pre-drained of their interiors for this purpose, although some recipes use full eggs.
As last week, they are invited to come up with recipes on paper for review by the professor before trying anything. Known possible results available with this set of ingredients include wooden, decorated eggs nested inside the shell; a shellful of small sharp teeth (the commonest source of hens' teeth, as they have encountered once or twice in potions); successively smaller eggshells the smallest of which holds a blob of mercury; and a perfectly hardboiled egg which can be arbitrarily flavored by exact details of the process. But they are welcome to aim at another result as long as their procedure looks safe.
"Well, alchemy seems to be the only known way to break otherwise inviolable laws of magic."
"Yeah, they are." Pause. "And now my brain's thinking of experimentation with new kinds of wands."
"New cores, combination cores, new woods, non-woods. I wonder if a chip of the philosopher's stone could be a wand core."
"Yeah! And it could be that they just don't work, but it could also be that attitudes like Mr. Ollivander's are widespread enough that people haven't tried, or not enough."
"I mean, if the effects people ascribe to them are really true, it could even make sense that they react to each other or something, but." Shrug.
She shrugs. "I dunno. Maybe if there was a way to find a Dementor in a supervised, safe environment, you could try patronusing it."
"Right. Yeah. They did." She shakes her head. "What in Merlin's name were they thinking..."
"Probably that if they couldn't throw them prisoners to feed off they'd have to have people casting Patronuses constantly to keep them penned in."
"Oh no. What a disaster. They would have to hire some people to keep casting a spell. Probably the same people who hung around them all day before. Terrible. So awful. Really, the best idea is to then let them loose so they can eat unsuspecting people. Muggles can't even see them."
"I don't think anybody actually did hang around all day before, except the prisoners. And you'd need a lot of Patronuses to cut off escape for that many Dementors. They might literally not have had the people to do it. But they should've got them."
"Yes, they should. I wonder if the fact that we don't hear fairly often about random Dementor attacks means that they're mostly preying on muggles."