it hatches while you hold it
An egg costs 0.0005 ETH and goes in the nest with your name on it. You don't hatch it. Every 10 minutes one egg in the nest cracks open by itself, and the ones that have waited longest go first.
What comes out is drawn from a bag of 500: 425 chicks, 60 ducks, 12 owls and 3 dragons. The sign shows what's still in the bag, so every chick that hatches makes the next egg likelier to be something better. The payout goes to whoever owns the egg.
Can't wait? Tap your egg and name a price. Anyone can buy an unhatched egg; the seller keeps 95%. Holders' eggs hatch before everyone else's.
Day 3 is a warm spell: an egg every 2 minutes for an hour. On day 7 whoever hatched the most takes what's left. Each egg sends 80% to the pot and 20% to buying $EGG and burning it.