Grease a 9x13-inch baking pan with cooking spray. Line with parchment paper, leaving an overhang on all 4 sides. Coat parchment paper with 1/4 cup confectioners' sugar.
Combine 1/2 cup water and gelatin in the bowl of a stand mixer fitted with the paddle attachment; let stand until gelatin softens, about 5 minutes.
Combine remaining 1/2 cup water, sugar, corn syrup, and salt in a saucepan. Bring to a boil, stirring constantly. Clip a candy thermometer to the side of the saucepan. Cook syrup over medium-high heat, stirring occasionally, until it registers 238 degrees F (114 degrees C) on the thermometer, 5 to 10 minutes.
Turn the stand mixer on low speed; pour half of the sugar syrup into the softened gelatin. Increase speed to medium and pour in remaining syrup. Increase speed to high and beat until marshmallow mixture is thick and creamy, 10 to 15 minutes. Beat in vanilla extract.
Scrape marshmallow mixture into the prepared pan and spread it to the edges. Smooth the top with a rubber spatula. Sprinkle 1/4 cup confectioners' on top. Let stand until firm and dry, 8 hours to overnight.
Lift marshmallow mixture out of the pan using the overhanging parchment. Transfer to a cutting board. Make a single cut through the center. Make 8 cuts evenly along the length and width to form 64 rectangles. Dip cut sides in remaining confectioners' sugar to prevent sticking.
Layer marshmallows in an airtight container, covering each layer with parchment paper or plastic wrap.