Bring water, vinegar, beet, shallot, sugar and a pinch of salt to a boil in a 2-quart saucepan. Simmer it, covered, until beet is tender, about 20 minutes.
Transfer the beet mixture to a container and cool it completely, uncovered.
Add the eggs to the mixture, submerging them entirely in the liquid. Refrigerate the eggs for at least 2 hours, and up to 6 hours, allowing the eggs to marinate while gently stirring them once or twice.
Remove the eggs from the beet mixture and pat them dry (discard the beet mixture). Halve the eggs lengthwise and remove the yolks.
Mash the yolks with the mayonnaise, mustard, sweet relish, and salt and pepper.
Divide the mixture among the egg whites. Sprinkle with a light dusting of paprika.
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