Buttered eggs are a specialty in Cork, Ireland.
Absolutely freshly-laid eggs straight from the nest are rolled in melted butter. The melted butter forms a protective seal over the shell, extending the storage life of the eggs in the absence of refrigeration, or even if they are refrigerated. The butter also permeates through the shell and influences the flavour of the egg inside.
You won’t see clumps of butter on the egg shells; it appears as a glaze.
Buttered eggs is also the name of a dish; see buttered eggs (2).