Brush a 9-by-13-inch glass baking dish with oil. Line with parchment, allowing a 2-inch overhang on the long sides. Brush parchment with oil; set aside.

Put granulated sugar, corn syrup, salt, and 3/4 cup water into a medium saucepan. Bring to a boil over high heat, stirring to dissolve sugar. Cook, without stirring, until mixture registers 238 degrees on a candy thermometer, about 9 minutes.

Meanwhile, put 3/4 cup cold water into the bowl of an electric mixer; sprinkle with gelatin. Let soften 5 minutes.

Attach bowl with gelatin to mixer fitted with the whisk attachment. With mixer on low speed, beat hot syrup into gelatin mixture. Gradually raise speed to high; beat until mixture is very stiff, about 12 minutes. Beat in vanilla. Pour into prepared dish, and smooth with an offset spatula. Set aside, uncovered, until firm, about 3 hours.

Sift 1 cup confectioners' sugar onto a work surface. Unmold marshmallow onto confectioners' sugar; remove parchment. Lightly brush a sharp knife with oil, then cut marshmallow into 2-inch squares. Sift remaining 1/2 cup confectioners' sugar into a small bowl, and roll each marshmallow in the sugar to coat. Marshmallows can be stored in an airtight container up to 3 days.