Soften the gelatin in 3/4 cup of the water in a small saucepan. Heat the gelatin slowly to fully dissolve it, but don't let it fully boil. Stir in the vanilla.
In a saucepan, combine the sugar, the corn syrup, and the remaining 3/4 cup water. Set over medium heat and stir until the sugar dissolves. Increase the heat and bring the mixture to a boil. Set a candy thermometer in the pan and, without stirring, let the mixture boil until it reaches 250° to 260°F (hard-ball stage). While the sugar mixture is boiling, beat the egg whites in the large bowl of a standing mixer fitted with the whisk attachment (or on high speed with a hand-held mixer) until they form medium-stiff peaks.
When the sugar syrup reaches temperature, whisk in the dissolved gelatin. Turn the mixer back on to the highest speed. Slowly pour the hot sugar syrup into the whisked egg whites, avoiding the whisk. The whites will get liquidy at first but will eventually thicken. Continue beating until the marshmallow mixture thickens enough to hold its shape, about 5 minutes; it will still be slightly warm.
Lightly oil two 8-inch-square pans and then dust them with some of the cornstarch and confectioners' sugar mixture. Divide the marshmallow mixture between the two pans, smooth it as flat as you can with a spatula, and let it sit uncovered until it's springy and firm, 4 to 6 hours.
Using your fingers or a sharp knife, loosen the marshmallows from the sides of the pans. Dust a long sheet of waxed paper with more of the cornstarch mixture. Flip the marshmallows onto the paper. Dust the surface with more of the cornstarch mixture and let the marshmallows sit for about 2 hours.
With a sharp knife or scissors, cut the marshmallows into even strips about 1-1/2 inches wide and then into squares. The marshmallows probably won't stick to the knife or scissors, but if they do, try oiling the blade.