Generously grease bottom and sides of 11x7-inch (2-quart) glass baking dish with butter; sprinkle with 2 tablespoons of the powdered sugar. Pour 1/2 cup cold water into bowl of heavy-duty mixer fitted with paddle or whisk attachment. Sprinkle gelatin over water; set aside.
In 2-quart heavy saucepan, heat granulated sugar, corn syrup, salt and 1/2 cup water over low heat, stirring constantly, until sugar is dissolved. Increase heat to medium-low and heat syrup to boiling. Boil without stirring until syrup reaches 240degreesF on candy thermometer, about 15 to 20 minutes; remove from heat.
With mixer running at low speed, slowly pour hot syrup into gelatin mixture in thin stream down side of bowl (avoid pouring syrup onto paddle or whisk, as it may splash). Gradually increase speed to medium-high; beat until mixture is white and has almost tripled in volume, about 8 to 10 minutes. Add peppermint extract and beat to blend, about 30 seconds longer. Randomly drop four drops of red or green food color into marshmallow mixture in bowl; fold 3 to 4 times to swirl color.
Pour into baking dish. Smooth top with damp rubber spatula. Drop red or green food color randomly on top of marshmallow mixture. Using toothpick, pull food color through marshmallow mixture to create swirl pattern over top. Let stand uncovered at least 4 hours.
Sprinkle cutting board with about 1 tablespoon of the 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. Spray sharp knife with cooking spray. Cut marshmallows 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.