The Egg Timer is an important item in Temtem if you are breeding. It lets you know how long you have before an egg hatches. To get it, you'll need to complete a special quest on Tucma, the third island in the game.
Where can I find the Temtem Connoisseur quest?
Go to Tucma, the third island, and then to Quetzal (the big city).
In the building to the right of the airship terminal in Quetzal, you'll meet a man named The Temtem Connoisseur. He likes to collect specific Temtems, and by talking to him you will be able to begin the quest.
The man is on the bench inside the building. He will ask you to raise a Bigu to complete his collection.
How to you raise a Bigu
Bigu is the "baby" form of the Temtem called Babawa. The only way to obtain a Bigu is to breed at least one Babawa with another Temtem.
Where can I find Babawa
To find a Babawa and breed it, you have to go back to the second island, in Omnimesia.
Breed Babawa to get Bigu
Once you have obtained your Babawa, return to Upper Omnimesis, in Mokupuni, to access the breeding center. (see map). Inside, a machine allows you to change the Temtem present in your team.
To obtain a Bigu, you have to give to the Breeding Center: a Babawa plus another Temtem of Grass or Water type. The breeding takes about 10 minutes.
You will then get an egg. You will have to put it in your team and travel a few miles to hatch it.
Go back to the Connoisseur
Once your Bigu is ready, return to Tucma, to Quetzal, show it to the Connoisseur and collect your Egg Timer.