Here is the American Egg Board’s tried and true method for cooking hard-boiled eggs:

Place eggs in a saucepan large enough to hold them in a single layer. Add cold water to cover the eggs by 1 inch. Heat over high heat just to boiling. Remove the pan from the burner. Cover the pan.

Let the eggs stand in hot water about 12 minutes for large eggs (9 minutes for medium eggs; 15 minutes for extra large).

Drain the eggs immediately and serve warm. Or cool them completely under cold running water or in bowl of ice water, then refrigerate.