Combine the 1/4 cup water, the corn syrup, and the sugar in a saucepan fitted with a candy thermometer. Bring to a boil and cook to "soft-ball" stage, or about 235 degrees F.
Meanwhile, in a standing mixer fitted with a whisk, whip the egg whites until soft peaks form. Sprinkle the gelatin over the 2 tablespoons water and let dissolve. When the syrup reaches 235 degrees F, remove it from the heat, add the gelatin, and mix. Pour the syrup into the whipped egg whites. Add the vanilla and continue whipping until stiff.
Transfer the mixture to a pastry bag fitted with a round tip. Pipe a "kiss" of marshmallow onto half of the cookies, and top with the rest to make sandwiches. Let set at room temperature for 2 hours.