Any battle (excluding Vortex) can drop materials for mogcakes. It's recommended to do exploration since they yield highest number of battles per energy. Higher energy dungeon or exploration will drop more material. Anything from Dirnado like [[Dwarves' Forge/Exploration]] or [[Maranda Coast/Exploration]] should be good. We don't know if [[Locke]] increases drop rate or not, but it never hurt to bring him in.
All recipes produces same amount of mogcakes in the same timeframe, though purchasable recipes have slightly lower material cost. If you are planning to buy all important items from King Mog, it's '''highly recommended''' to buy a rare or premium recipe as well as additional crafting slot. You can still craft and trade with King Mog after the event period is over but before the claim period is over. The claim period lasts 21 days, and thus you can craft up to '''90,000 mogcakes''' per crafting slot.