In FFXIV, the Sunset Carrot is a crafting material used to create certain food items as a Culinarian crafting class. It can be purchased from any Scrip Exchange NPC (such as the one in the Studium in Old Sharlayan). Each costs 15 Purple Crafters’ Scrips apiece.
The Culinarian recipes that require Sunset Carrots also require the player to first purchase (and use) a copy of Master Culinarian X — which can also be purchased at the Scrip Exchange for White Crafters’ Scrips.
Sunset Carrots were introduced in Patch 6.2 during Endwalker.