In a stand mixer fitted with the whisk attachment, add the gelatin to 1/2 cup of the cold water.
Combine the remaining 1/2 cup water, the granulated sugar, corn syrup, and salt in a 2-quart saucepan. Do not stir the mixture while it cooks — it will create crystallization and we don’t want that. Place over medium-high heat, cover, and allow to cook for 3 to 4 minutes. Uncover and continue to cook until the mixture reaches 240°F on a digital instant-read thermometer, approximately 8 minutes. When the mixture reaches this temperature, immediately remove from the heat.
Turn the mixer to low speed and, while running, slowly pour the sugar syrup down the side of the bowl into the gelatin mixture. Once you have added all of the syrup, increase the speed to high. Continue to whip until the mixture becomes very thick and is lukewarm, 12 to 15 minutes. Add the vanilla during the last minute of whipping. While the mixture is whipping, prepare the pan as follows.
For regular marshmallows: Combine the confectioners' sugar and cornstarch in a small bowl. Lightly spray a 9-by-13-inch metal baking pan with nonstick cooking spray. Add the sugar and cornstarch mixture and move around to completely coat the bottom and sides of the pan. Return the remaining mixture to the bowl for later use.
When ready, pour the mixture into the prepared pan, using a lightly oiled spatula for spreading evenly into the pan. Dust the top with enough of the remaining sugar and cornstarch mixture to lightly cover. Reserve the rest for later. Allow the marshmallows to sit uncovered for at least 4 hours and up to overnight.
Turn the marshmallows out onto a cutting board and cut into 1-inch squares using a pizza wheel or knife dusted with the confectioners' sugar mixture. Once cut, lightly dust all sides of each marshmallow with the remaining mixture, using additional if necessary. Store in an airtight container for up to 3 weeks.
For miniature marshmallows: Combine the confectioners’ sugar and cornstarch in a small bowl. Line 4 half-sheet pans with parchment paper, spray the paper with nonstick spray and dust with the confectioners’ sugar mixture.
When ready, scoop the mixture into a piping bag fitted with a 1/2-inch-round piping tip. Pipe the mixture onto the prepared sheet pans lengthwise, leaving about 1 inch between each strip. Sprinkle the tops with enough of the remaining cornstarch and sugar mixture to lightly cover. Let the strips set for 4 hours or up to overnight.
Cut the strips into 1/2-inch pieces using a pizza wheel or scissors dusted with the confectioners’ sugar mixture. Once cut, lightly dust all sides of each marshmallow with the remaining sugar mixture and store in an airtight container for up to 1 week.