Line a 13x9x2-inch baking pan with plastic wrap or line bottom of pan with waxed paper or parchment paper. Coat the plastic or paper with nonstick cooking spray; set pan aside.
In a large metal or heatproof bowl, sprinkle gelatin over 1/2 cup of the cold water; set aside.
In a 2-quart heavy saucepan, stir together remaining 1/4 cup water, 1-3/4 cups of the sugar, and the corn syrup until combined. Bring to boiling over medium-high heat. Clip a candy thermometer to the side of the saucepan. Cook, without stirring, over medium-high heat until thermometer registers 260 degrees F, hard-ball stage (12 to 15 minutes total). Remove from heat; pour over gelatin mixture in bowl and stir well to combine (mixture will foam and bubble up in the bowl).
Meanwhile, in a clean mixing bowl, beat the egg whites, vanilla, and salt with an electric mixer on high speed until foamy. Gradually add remaining 1/4 cup sugar, 1 tablespoon at a time, until stiff peaks form (tips stand straight). With the mixer running on high speed, gradually add gelatin mixture to egg white mixture, beating for 5 to 7 minutes or until thick (the consistency of thick, pourable cake batter). Quickly and gently spread marshmallow mixture into prepared pan. Coat another piece of plastic wrap with nonstick coating; place, coated side down, over marshmallow mixture in pan. Chill at least 5 hours or until firm.
In a bowl, combine powdered sugar and cornstarch; sprinkle about 1/4 of the mixture evenly onto a large cutting board. Remove plastic wrap from top of marshmallows. Loosen sides of marshmallow mixture if necessary and carefully invert onto the cutting board. Remove plastic wrap or paper. Sprinkle top with some of the remaining powdered sugar mixture. Cut marshmallows into 1-inch squares. Place squares, about one-third at a time, in a large resealable plastic bag. Add remaining powdered sugar mixture; seal bag and shake to coat all sides of marshmallows with powdered sugar mixture.