Generously grease bottom and sides of 11x7-inch (2-quart) glass baking dish with butter; dust with 1 tablespoon of the powdered sugar. In bowl of stand mixer, sprinkle gelatin over 1/2 cup cold water to soften; set aside.
In 2-quart saucepan, heat granulated sugar, corn syrup, salt and 1/2 cup water over low heat, stirring constantly, until sugar is dissolved. Heat to boiling; cook without stirring about 30 minutes to 240°F on candy thermometer or until small amount of mixture dropped into cup of very cold water forms a ball that holds its shape but is pliable; remove from heat.
Slowly pour syrup into softened gelatin while beating on low speed. Increase speed to high; beat 8 to 10 minutes or until mixture is white and has almost tripled in volume. Add peppermint extract; beat on high speed 1 minute. Pour into baking dish, patting lightly with wet hands. Drop food color randomly onto top of marshmallow mixture. Pull table knife through food color to create swirl pattern over top. Let stand uncovered at least 8 hours or overnight.
Dust cutting board with about 1 tablespoon powdered sugar. Place remaining powdered sugar in small bowl. To remove marshmallow mixture, loosen sides from dish and gently lift in one piece onto cutting board. Using sharp knife greased with butter, cut into 1-inch squares (11 rows by 7 rows). Dust bottom and sides of each marshmallow by dipping into bowl of powdered sugar. Store in airtight container at room temperature up to 3 weeks.