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.
"Okay, different tacks, um. I'm a social person, I like people, I'm doing the networking thing, right, so it makes sense the kind of memory that makes it work for me involves that. You're different, you're introverted, more studious. What's the equivalent thing...?"
"Uh... Figuring things out? Learning new things? Getting things you want? Achieving goals? That's a you thing, achieving goals."
"Maybe the thought of achieving them, then? Uh, getting into the alchemy thing may count?"
She tries.
Mist. Maybe with a little more definition around the edges, but not enough edges to be a real shape.
"Okay, that's progress! Maybe after we make the panacea or the philosopher's stone that'll work."
"Most people don't cast one, full stop. But that was a worst-case scenario, we just need something significant enough? Maybe? I mean this is just a theory, your mist getting a little bit better defined is definitely not enough."
"...okay, um, what does 'getting into alchemy class' have in general that just the pearl doesn't have in specific?"
"Okay... so try to focus on that in specific, maybe? Visualise getting a legendary alchemic rock."
"Okay, yes, that looks like we're on the right track. Visualise... getting the Elixir of Life? Scaling it up? Everyone immortal and immune to disease?"
Slightly defined mist.
"Yeah, but we're on the right track, I think. Except, we're looking at lots of yesses, how do we falsify this hypothesis?"
"Yeah, but it's something to do with legendary alchemical things, and making everyone immortal and stuff, which points to a general 'getting what you want,' but it also could be a specific 'making people immortal' thing."
"Okay, so, as long as I'm imagining things..." Incantation. Mist. "It is not just a getting what I want thing."
"Yeah, okay, similar scale. So it's something about making people immortal? Is there a difference between a panacea-mist and a philosopher's stone-mist?"