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To hard cook eggs: Gently place eggs in single layer in medium saucepan. Add enough cold water to cover eggs with 1 inch of water. Bring just to boil on medium-high heat. Remove from heat. Cover and let stand about 12 minutes for large eggs. (Adjust time up or down by 3 minutes for each size larger or smaller.) After 12 minutes, pour off the hot water and rapidly cool eggs by running them under cold water (or place in ice water) until completely cooled. Refrigerate and use within a week.
For creamier deviled eggs: Stir additional 1 to 2 tablespoons mayonnaise into yolk mixture.