Make foil sling for 13 by 9‑inch baking pan by folding 2 long sheets of aluminum foil; first sheet should be 13 inches wide and second sheet should be 9 inches wide. Lay sheets of foil in pan perpendicular to each other, with extra foil hanging over edges of pan. Push foil into corners and up sides of pan, smoothing foil flush to pan. Spray pan with vegetable oil spray. Whisk confectioners’ sugar and cornstarch in small bowl; set aside.
Pour 1/2 cup water into bowl of stand mixer fitted with whisk. Sprinkle gelatin over water. Let stand until gelatin becomes very firm, about 15 minutes.
Meanwhile, combine remaining 1/2 cup water and corn syrup in medium saucepan. Pour granulated sugar and salt into center of saucepan (do not let sugar hit saucepan sides). Bring to boil over medium-high heat and cook, gently swirling saucepan, until sugar has dissolved completely and mixture registers 240 degrees, 6 to 8 minutes.
Turn mixer speed to low and carefully pour hot syrup into gelatin mixture, avoiding whisk and bowl. Gradually increase speed to high and whip until mixture is very thick and stiff and coats whisk, 10 to 12 minutes, scraping down bowl as needed. Add vanilla and mix until incorporated, about 15 seconds.
Working quickly, scrape mixture evenly into prepared pan using greased rubber spatula and smooth top. Sift 2 tablespoons confectioners’ sugar mixture over pan. Cover and let sit overnight at room temperature until firm.
Lightly dust cutting board with 2 tablespoons confectioners’ sugar mixture and lightly coat chef’s knife with oil spray. Turn marshmallow slab out onto cutting board and peel off foil. Sift 2 tablespoons confectioners’ sugar mixture over slab. Cut into 1‑inch-wide strips, then cut crosswise into 1-inch squares. Working with 3 or 4 marshmallows at a time, toss marshmallows in bowl with remaining confectioners’ sugar mixture, then toss in fine-mesh strainer to remove excess powder. Marshmallows can be stored in zipper-lock bag or airtight container for up to 2 weeks.