Mongo's Hat is crafted with Tailoring in Mabinogi (hat). This page lists everything needed to make it, checked against the game as it exists today. The interactive planner lives in the Tailoring Companion.
How to get the sewing pattern
The sewing pattern is rank A and survives 30 uses; every crafting attempt spends one use.
Nora: Buy it from Nora at the inn in Tir Chonaill. (11,250 gold). No special requirement, just enough gold.
Granites: Buy a pattern from Granites's general store in southwest Filia. (11,250 gold). Elf-allied humans get a discount, other races pay full price.
Sources that no longer exist: Beginner Gachapon (2012).
Materials
Expect about 2 attempts per finished item (50% average progress per attempt). Per-attempt materials are consumed on every attempt; finishing materials once per item.
Each attempt uses
1× Finest Fabric — weaving on a loom, higher rank
1× Common Silk — buy from Seal Merchant (Mondays) for seals, or weave rank F
Finishing (once per item)
1× Finest Finishing Thread — Buy directly from any general goods NPC shop.