Use this method for preparing hard-boiled eggs when you live above 5,000 feet.
Place eggs in saucepan large enough to hold them in a single layer. Add cold water to cover eggs by 1 inch. Cover pan and heat on high just to boiling. Reduce heat and simmer for 5 minutes.
Remove pan from burner. Let eggs stand, covered, 15 minutes for large eggs/20 minutes for extra-large eggs.
Drain immediately and serve warm, or cool completely under cold running water or in a bowl of ice water, then refrigerate.
From Colorado State University — Colorado Farm to Table Food Safety website.