Beat the eggs with the mustard in a medium-size bowl.
Beat in the lemon juice, shallot, and tarragon.
Place the bowl on a pot holder or a folded cloth so it is stable. Then, beating constantly with a whisk, pour in about half the oil in a very thin stream. The mixture should thicken like a mayonnaise.
Once about half the oil has been added, beat the mixture for a minute or so until it begins to lighten in color. Then add the remaining oil in a stream still beating constantly.
When all the oil has been added, continue beating for a few seconds, then season with salt and pepper. Stir in the water.