Put egg whites in the large bowl of a mixer. Stir 1/4 cup cool water and the gelatin in a glass measuring cup; let stand to soften while you make syrup.
Stir granulated sugar, corn syrup, and 2/3 cup water in a small saucepan. Insert a candy thermometer. Boil over high heat, tipping pan occasionally to cover thermometer bulb, until 240°, 6 to 10 minutes. Immediately microwave gelatin mixture until it dissolves, about 30 seconds, and also beat egg whites on high speed into soft peaks. Stir steaming gelatin into hot syrup.
Pour about 3 tbsp. syrup at a time into egg whites (stop mixer if needed to keep syrup from flying up onto inside of bowl), beating on high speed 20 to 30 seconds after each addition. Continue to beat until meringue holds soft peaks and underside of bowl is completely cool, 5 to 7 minutes. Beat in vanilla.
Spread marshmallow mixture in coated pan, using an oiled metal spatula. Let stand until firm enough to cut, 2 hours. Sprinkle with 2 tbsp. reserved cornstarch mixture, then cut with a pizza cutter or scissors into 12 rectangles to fit graham crackers; you may have leftover marshmallows.
Toss marshmallows in bowl with remaining cornstarch mixture to coat.
Make ahead: Up to 4 days, chilled airtight with plastic wrap between layers; coat with more cornstarch mixture if they get sticky.