Put eggs in a pot of cold water. Heat to boiling; lower heat. Simmer for 15 minutes. Pour out the water, add ice cubes and cold running water. Peel the eggs. Chop them, or slice them twice with an egg slicer, slicing in two different directions.
While the eggs are cooking, chop the potatoes into half-inch cubes. Put in a pot. Add enough cold water to cover the potatoes. Heat to boiling. Boil 10 to 12 minutes, until the potatoes pierce easily with a fork. Drain.
While the potatoes are boiling, make the beannaise.
Put all of the ingredients into a big bowl. Stir thoroughly.
Serve right away, or chill it for several hours before serving.