Lightly coat a 9-by-9-inch baking pan with cooking spray. Line pan with parchment paper, leaving a 4-inch overhang on all sides. Coat parchment with cooking spray; set aside.
In the bowl of an electric mixer fitted with the whisk attachment, sprinkle gelatin over 1/2 cup cold water; let stand for 10 minutes. In a medium saucepan, combine sugar, corn syrup, and remaining 1/4 cup water. Place saucepan over medium-high heat and bring to a boil; boil rapidly for 1 minute. Remove from heat, and, with the mixer on high, slowly pour the boiling syrup down the side of the mixer bowl into gelatin mixture. Add salt and continue mixing for 12 minutes.
Add vanilla extract; mix until well combined. Spray a rubber spatula or your hands with cooking spray. Spread gelatin mixture evenly into pan using prepared spatula or your hands. Fold parchment overhangs over to cover. Let stand for 2 hours.
Carefully remove marshmallows from pan. Sprinkle work surface with confectioners' sugar. Remove parchment and place on top of sugar. Cut marshmallows into 16 pieces using a sharp knife sprayed with cooking spray. Place confectioners' sugar in a large bowl. Working in batches, add marshmallows to bowl and toss to coat.