3 cups water
1 cup distilled white vinegar
1 small beet, peeled and sliced
1 small shallot, sliced
1 teaspoon sugar
1 bay leaf
12 hard-boiled large eggs, peeled
1/3 cup mayonnaise
1 tablespoon grainy mustard
1 teaspoon Siracha Hot Chili Sauce
Sweet Paprika, to taste
Bring water, vinegar, beet, shallot, sugar, bay leaf, and 1/2 teaspoon salt to a boil in a 2-quart saucepan, then simmer, covered, until beet is tender, about 20 minutes. Cool completely, uncovered. Put beet mixture in a container with eggs and marinate, chilled, gently stirring once or twice, for 2-6 hours.
Remove eggs from beet mixture and pat dry (discard beet mixture). Cut in half lengthwise and remove yolks. Mash yolks with mayonnaise, mustard, and Siracha sauce. Season with salt and pepper, then divide among egg whites and sprinkle with paprika.