Place the eggs in a large pot. Add enough water to the pot to completely cover the eggs by ½ inch. Bring the water to a boil. Boil the eggs for 12 minutes.
Drain the hot water from the pot. Cover the eggs (still in the pot) in cold water. Let the eggs rest for 2 minutes. Drain the water and repeat this step until the eggs are completely cool.
Peel the shell from the eggs. Cut the eggs in half, lengthwise.
Remove the yolks from the egg whites. Discard one of the egg yolks (2 halves). Add the remaining 5 egg yolks to a small bowl. Mash the yolks with the back of a fork until there are no large lumps. Add Greek yogurt, mustard, vinegar, sugar, salt, and pepper to the bowl. Stir until well combined and smooth.
Transfer the filling to a small plastic food storage bag. Snip one of the corners off of the bag. Squeeze the filling through hole into the egg whites. Or, spoon the filling into the egg whites.