Step 1: Grease a 9-by-13-by-2-inch pan with vegetable shortening: Dab a little bit of shortening on a paper towel and rub it into the sides and bottom of the pan. Set aside.
Step 2: Put the gelatin sheets in a medium or large heatproof bowl, fill the bowl with very cold water, and set aside; add a few ice cubes to keep the water cold.
Step 3: In a medium saucepan, gently stir together the sugar, ½ cup of the corn syrup, and ½ cup water. Be careful not to splash the ingredients onto the sides of the pan. Put the saucepan over medium-high heat and clip a candy thermometer to the side of the saucepan.
Step 4: Fill a medium saucepan halfway with water and place on the stove over medium-low heat.
Step 5: Put the remaining ½ cup corn syrup in the bowl of an electric mixer fitted with the whisk attachment. Set aside.
Step 6: Return to the saucepan of water and check the temperature. When the temperature reaches 220 degrees F., drain the water from the bowl of gelatin and give the gelatin sheets a quick wringing out. Place the bowl of gelatin over the saucepan of simmering water and stir the gelatin sheets with a heatproof spatula until the gelatin is completely melted. Remove the bowl from the pan. Turn the mixer on low speed and slowly pour the melted gelatin into the corn syrup. Keep the mixer on low.
Step 7: Bring the sugar mixture to the soft ball stage on the candy thermometer (235 to 240 degrees F.), then remove from the heat. Take out the candy thermometer. Turn the mixer up to medium for 1 minute, then slowly pour the sugar mixture into the gelatin mixture. When all of the sugar mixture has been added, turn the mixer to medium-high and beat for about 5 minutes. The marshmallow mixture will begin to turn white and fluffy. Add the vanilla and salt and turn the mixer up to its highest setting for another minute.
Step 8: Working very quickly, pour the marshmallows into the prepared pan. Use an offset spatula to spread out the mixture evenly. Sprinkle with a bit of sifted confectioners’ sugar and let sit for about 6 hours.
Step 9: Use a knife to loosen the marshmallow from the edges of the pan, and use your hands to pull the marshmallow (it will come out in one gigantic piece) out of the pan and onto a flat surface lightly dusted with confectioners’ sugar. Place the ½ cup confectioners’ sugar in a small bowl.
Step 10: Use a chef’s knife to cut the marshmallows into a 6-by-8 grid. Roll each marshmallow in confectioners’ sugar. The marshmallows will keep in an airtight container for up to 1 week. Serve with hot chocolate or eat as is.