Grease a 9×13-inch baking pan with cooking spray; dust with 1 tablespoon confectioners’ sugar.
Pour 1/2 cup lukewarm water into the bowl of a stand mixer fitted with the whisk attachment. Sprinkle 2 tablespoons plus 3/4 teaspoon gelatin on top; stir briefly to combine.
Combine sugar, corn syrup, 1/2 cup water, and salt in a large saucepan. Cook over medium-low heat, stirring constantly, until sugar dissolves, about 5 minutes. Increase heat and cook, stirring constantly, until mixture registers 250 degrees F (121 degrees C) on a candy thermometer, about 5 minutes. Remove from heat.
Turn the mixer on low speed and, while running, pour the hot sugar syrup in a thin stream down the side of the bowl. Increase speed to high and beat until marshmallow mixture is lukewarm and triples in volume, about 15 minutes. Mix vanilla extract.
Pour marshmallow mixture into the prepared pan. Press into an even layer with wet hands. Dust with 2 tablespoons confectioners’ sugar. Let stand at room temperature, uncovered, until set, 8 hours to overnight.
Cut marshmallow mixture into desired shapes and toss in remaining 2 tablespoons confectioners’ sugar.