In the bowl of a standing mixer fitted with a whisk, soften the gelatin in 1/2 cup plus 2 teaspoons cold water for 30 minutes.
In a small, heavy saucepan, add the granulated sugar, 1/2 cup water, the corn syrup and salt; cook over medium-low heat, stirring gently, until dissolved. Increase the heat to high and cook, without stirring, until the syrup registers 240 degrees on a candy thermometer, 12 to 15 minutes. Remove from the heat.
Line a 13-by-18-inch rimmed baking sheet with plastic wrap. Spray with cooking spray. Beat the hot syrup into the gelatin mixture on low speed. Increase the speed to high and beat until the marshmallow is doubled, about 15 minutes. Beat in the vanilla. Spray a spatula with cooking spray and spread the marshmallow evenly onto the baking sheet. Lightly spray with cooking spray, cover with plastic wrap and refrigerate for 5 hours or overnight.
In a double boiler, melt half of the chocolate, stirring, until smooth. Add the remaining chocolate, remove from the heat and stir in the oil until smooth.
Line a cooling rack with wax paper. Holding by a corner, dip 20 graham cracker halves into the chocolate to coat. Place the grahams on the wax paper; refrigerate to set, about 30 minutes.
Cut the marshmallow into 2-inch squares and toss with the confectioners' sugar in a large bowl.
Stack a plain graham, marshmallow square, chocolate-coated graham, another marshmallow and another plain graham. Makes 20 stacks.