Combine 2 quarts water with the food coloring in a saucepot. Gently place the eggs in the water. Set over high heat and bring to a boil. Once boiling, cover, remove from the heat and set aside for 10 minutes.
Remove the eggs from the water and place on paper towels. Lightly tap the eggs with a spoon to crack the shells. Transfer the eggs to a container with some ice and pour the cooking liquid over the eggs. Chill until completely cold.
Gently peel the eggs. Slice a thin sliver off the narrow end of each egg so the eggs can stand upright when plated. Cut off the wide end of each egg and use a small spoon or the tip of a butter knife to gently scoop out the egg yolk into a mini food processor or mixing bowl. Process the yolks (or smash with a fork). Add the mayonnaise, vinegar, salt and pepper and mix until smooth. Spoon or pipe the yolk filling into the egg white cups.
Place the egg white cups upright on a serving platter. Sprinkle a pinch of paprika on top of the yolks. Finish each egg with a sprig of fresh dill. Serve chilled.