Generously dust rectangular baking dish, 11x7 1/2x2 inches, with powdered sugar. Sprinkle gelatin on 1/2 cup cold water in large bowl to soften; set aside.
Heat granulated sugar, corn syrup, salt and 1/2 cup water in 2-quart saucepan over low heat, stirring constantly, until sugar is dissolved. Heat to boiling; cook without stirring to 250º on candy thermometer or until small amount of mixture dropped into very cold water forms a ball that holds its shape but is pliable; remove from heat.
Slowly pour syrup into softened gelatin while beating with electric mixer on high speed. Beat on high speed until mixture is white and has almost tripled in volume. Add peppermint extract; beat on high speed 1 minute. Pour into pan. Sprinkle with powdered sugar, patting lightly with hands. Let stand uncovered at least 8 hours.
Turn pan upside down to remove marshmallow mixture onto board. Cut into shapes with miniature cookie cutters or knife dipped in water to keep from sticking. Store in airtight container at room temperature up to 3 weeks.