The Queen's Egg is the final egg in Mimic Egg Heist. It belongs to the Mimic Queen, the mother of all Mimics, and she sleeps curled around it on an island past the Rainbow. Steal it and make it home, and Aurum the Golden Mimic is yours.
Step 1: collect 15 key shards
The way to her island is locked. You need a key shard from every area, 15 in all, and you get an area's shard by bringing home a Legendary or better egg from that area.
Because you only see an egg's rarity once you grab it, this takes some runs. The rarities guide explains what counts as Legendary or better.
Step 2: through the Queen's Gate
With all 15 shards, the Queen's Gate at the far end of the Rainbow area lifts you onto her island.
Step 3: freeze when she looks
The Mimic Queen sleeps for a few seconds at a time, five to nine seconds. Then her eyes flutter for about a second as a warning, and then they stay open for a couple of seconds.
While her eyes are open, stand still. Move faster than a slow creep while she is looking and she throws you off the island. You can only take the egg while she is asleep.
So the rhythm is: move while she sleeps, freeze when her eyes flutter, wait until they shut, move again.
Step 4: run
The Queen's Egg is heavy. Carrying it, you walk at a bit more than half your normal speed. Every guardian on your way home wakes up, the whole server is told you have it, and any other player can bonk the egg off you.
Train your speed before you try this. The run home is the hard part.
The reward
Make it home and you get Aurum, the Golden Mimic, as a Divine Rainbow pet. Aurum only comes from the Queen's Egg.
You also get a crown tag, and every Queen's Egg you bring home gives you +5% income, up to ten times.