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.
"Do you remember anything more about it than just thinking that Dementors exist?"
"No, I mean, I must have been thinking something else about it but I don't recall what, that was all I was trying to concentrate on."
She closes her eyes and thinks. "Dementors exist, that's a bad thing, there must be some good out of them existing...? Or something good that can only happen if they exist..."
"Ermm... some goal of yours that involves Dementors? Something you can achieve, that has to do with immortality? Or power, maybe, if it's not immortality? Life, er, happiness..."
"...oh. Ooh. Okay, the book said Dementors are indestructible but who cares, we're recreating the philosopher's stone, we don't need to worry about rules. Try something like that? Dementors not existing? Maybe that's too abstract, but, erm, something to destroy Dementors? Like a superpowerful Patronus or something."
"I mean, the theme seems to be 'things magical society generally deems impossible' but. Maybe we could try to generate some memories related to that, somehow?"
"Yeah, I understand," she sighs. "But hey, at least we got something today, it won't be just random guessing."