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.
"Hmm, is that a mindset thing? If I try the same thing twice but wanting or thinking of different results do I get different things?"
He turns to Miranda. "So, as I was saying, this is deterministic so controlled trials aren't really all that necessary."
"Well I'd assumed it, and questioned it as soon as it came to my conscious attention and I turned out to be right, so!" He sticks out his tongue at her. "Besides, I'd totally have tried doing the same thing many times."
"I'm just not ascribing you quite the patience, when you could instead try radically different things."
"I can be patient! I can totally be patient! Just watch me patiently waiting for the paint to dry, here!"
"I'm not pretending!" he protests. "I just don't have a reason to be patient right now, but I could totally be patient for stuff I really wanted. Like making Slytherin nice again! Or making everyone immortal."
"Yes but both of those things are things I haven't even tried doing yet! The Slytherin thing is an ongoing project that will take yeeeears, and the immortal thing we won't even really try for a good long while."
The paint is dry! Except under the egg, so he turns it around and paints the rest. "And even if it is I still think I'd be able to do it."
"Do you think you wouldn't? If it turned out there weren't clever tricks like 'asking the Professor about it' available?"
"Hmm. No I'm not that thing. That's not what I meant with it. I meant the other thing."