Coat a 12-by-17-inch rimmed baking sheet with cooking spray; line with parchment paper. Spray parchment; set aside. Pour 1/3 cup cold water into the bowl of an electric mixer. Sprinkle with gelatin; let mixture soften, about 5 minutes.
Place sugar, corn syrup, salt, and 1/3 cup water in a medium saucepan. Cover; bring to a boil. Remove lid; cook, swirling pan occasionally, until syrup reaches 238 degrees (soft-ball stage) on a candy thermometer, about 5 minutes.
With mixer on low speed, whisk gelatin mixture, and slowly pour the syrup in a steady stream down the side of the bowl (to avoid splattering). Gradually raise speed to high; beat until mixture is thick, white, and has almost tripled in volume, about 12 minutes. Add vanilla, and beat 30 seconds to combine.
Pour mixture onto prepared baking sheet; smooth with an offset spatula. Let stand at room temperature, uncovered, until firm, at least 3 hours or overnight.
Coat a 1- or 2-inch snowflake-shaped cookie cutter with cooking spray to prevent it from sticking. Cut out as many individual marshmallows as possible; coat cutter with more spray as needed. Use marshmallows immediately or store in an airtight container at room temperature up to 1 week.
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