To hatch the Egg of Altasgracias, this is a multistep side quest. First you need to acquire 3 items (Black Grieving Veil, Torn Bridal Ribbon, & Three Melted Coins), place them on the offering plates to get the Egg of Deformity, offer that egg to a tree, then leave and come back to receive the hatched egg and the achievement.
First collect these 3 items:
1. Black Grieving Veil: can be found in the Desecrated Cistern.
2. Torn Bridal Ribbon: can be bought for 800 Tears from Candelaria (the shopkeeper lady) at the bottom of the Graveyard of the Peaks.
3. Three Melted Coins: can be found in Grievance Ascends and requires Blood Perpetuated in Sand relic to reach.
Once you have these 3 items, you need to place them on the plates in front of the hairy egg at the very bottom part of the map (at the bottom of Grievance Ascends). After placing them on the 3 plates, you'll receive the Egg of Deformity from Altasgracias. You then need to take this egg to the tree at the center of Mountains of the Endless Dusk. Once you offer the egg to the tree, you need to leave the screen and then come right back and the tree will give you the Hatched Egg of Deformity which unlocks this achievement.
With the Hatched Egg, you can take it to the Blessed Lord of the Salty Shores (the statue inside the shrine in Albero that teleports you to the snowy area where you get your items blessed) which will reward you with Three Gnarled Tongues relic (which grows branches in specific areas allowing you to reach previously inaccessible collectibles).
