Put the sugar, water, vinegar and butter into a cast iron saucepan and cook them over medium heat, stirring constantly until the sugar has dissolved. Bring the mixture to a boil, and boil it for 15 minutes. Test a teaspoonful of the boiling mixture in cold water; if it hardens at once (soft crack stage) remove the mixture from the heat.
Pour the bulk of the mixture into a buttered or oiled platter or marble slab, but retain a little in the saucepan, and keep the saucepan in a large pan filled with warm water to prevent the taffy mixture from hardening.
Grease your hands with butter and, as quickly as possible, pull the poured taffy while it is hot, adding a few drops of peppermint extract while pulling. Continue pulling until the taffy turns a creamy color, then pull it into long, flat strips about 1 inch wide.
Butter or oil the slab or dish again. Lay the taffy strips on the slab or dish. Pour the reserved taffy mixture in a thin stream onto the taffy strips to form a thin brown line along the center of each one. Cut the taffy into small pieces before it hardens. Makes about 3/4 pound.