Put eggs into a small pot; cover with water by 2 inches. Bring to a boil, cover, and remove from heat. Let stand for 8 minutes, drain, and run eggs under cold water to chill. Gently peel eggs and halve each lengthwise. Remove yolk from each egg half and transfer to a large bowl.

Arrange whites on a large platter, cover, and refrigerate. Mash yolks well with a fork, then add mayonnaise, vinegar, butter, mustard, and sugar. Season with salt and pepper to taste. Stir the egg yolk mixture until smooth. Using a small spoon, fill each egg white half with a rounded dollop of the egg yolk mixture. Alternatively, use a spatula to transfer the filling to a piping bag fitted with a ¹⁄₂-inch round or fluted tip; pipe a dollop into each egg white half. Loosely cover deviled eggs with plastic wrap and refrigerate until chilled. When ready to serve, sprinkle the eggs with scallions and paprika.