Line the bottom and sides of a 13- by 9-inch baking pan with plastic wrap; oil and then generously dust bottom and sides with some powdered sugar.
In the bowl of an electric mixer, place 1/3 cup cold water; sprinkle the gelatin over the surface.
In a heavy saucepan with a tight fitting lid, add sugar, corn syrup, salt, and remaining 1/3 cup water; stir to dissolve sugar. Cover the pan and place over moderately low heat. Remove the cover after 4 to 5 minutes. The steam will have caused any sugar crystals to dissolve and the syrup will be bubbling lightly. Increase the heat to high, insert a candy thermometer, and boil the syrup, without stirring, until it reaches 240 degrees F. Immediately remove from the heat.
Fit your electric mixer with the whisk attachment. slowly and carefully pour the syrup into the gelatin while the mixer is beating constantly at medium speed. When all of the syrup has been added, increase the speed to high and whip for approximately 10 minutes until the mixture is lukewarm very white, and the consistency of marshmallow cream. Add the vanilla extract toward the end of mixing.
Pour the marshmallow mixture into the prepared pan; smooth the top and sprinkle liberally with colored sugar of your choice. Let the pan stand, uncovered, at room temperature to dry. out. NOTE: Depending on the humidity, this may happen in several hours or take up to 8 hours. Generally speaking the longer you let it set up, the easier the marshmallow sheet will be to cut.
When ready to cut, invert the pan of marshmallow onto a clean cutting surface; remove the plastic wrap and coat the top with colored sugar (it should adhere easily).
Use cookie cutters to stamp out your peeps (or bunnies) and toss them in a bowl of sugar to coat the edges. If you find your cookie cutter getting sticky, was it and lightly coat with vegetable oil. With a toothpick apply a dot of chocolate to form an eye.
Store the marshmallow peeps in an airtight container.