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.
"Maybe? I'd meant that maybe it won't work for us until something horrible enough happens to us that we can 'truly appreciate' happiness or something," she says, air-quoting with her fingers. "That'd explain why some people manage to go from nothing to full corporeal Patronus when they see a Dementor."
"Sure, but we could still give it a whirl juxtaposing whatever we've got. ...I'm not very well equipped there either, but, still."
"I was planning on using free period to practise Charms, but I could do that around you two."
"I concentrate better when nobody's going 'expecto patronum' across the room repeatedly..."
"I wonder if the Patronuses pass through all the intervening space. We might alarm people wandering the halls if we did that."
"I think Spukhafte said if the target's too far the Patronus just disappears and appears there. I wonder if it does FTL, it'd be just like magic to do FTL without telling anyone."
"Well, I don't think we can easily test that on Earth unless one of us goes to Australia and we have hyper-accurate clocks that work around magic."
"I wonder if Patronuses can be used to collect information other than relaying messages. Like, 'Go to this address then return and tell me what you see.'"
"I feel like that's the sort of thing Professor Spukhafte would have known, unlike the FTL thing."
"That's true. ...I wonder if apparition is FTL. Actually I wonder what-all even goes on with apparition, from what I've read we go through some bizarre non-space to get there and it feels weird, so if there's some wormhole or something going on it'd totally be possible for it to be FTL, too."
"I think you can't Apparate all the way around the world in one go, so it'd be hard to tell if that was FTL at all."