In a medium heavy-bottomed saucepan over medium-high heat, stir together sugar, water, corn syrup and salt. Boil until the temperature reaches 238 degrees, also known as soft-ball stage.
Meanwhile, in the bowl of a stand mixer fitted with the whisk attachment, whip egg whites on medium speed until they hold soft peaks, about 2 minutes. With the mixer on medium speed, slowly pour half of the hot syrup into egg whites, beginning with just 1 to 2 tablespoons and gradually pouring in the rest (this method warms up the egg whites and prevents them from scrambling).
Immediately return saucepan with the remaining hot syrup to medium-high heat and bring to a boil; continue beating egg white mixture until thick, with the consistency of marshmallow creme. Mix in vanilla and vanilla bean paste. Turn off mixer.
When the syrup temperature reaches 275 degrees, also known as soft-crack stage, turn mixer on at medium speed and slowly add syrup to egg white mixture. Increase mixer speed to high and beat for 10 to 12 minutes, until the mixture is very thick and beginning to lose its gloss; the bowl should be cool except for the very bottom.
Scrape mixture into prepared pan and spread into the corners with lightly oiled hands or an offset spatula. Let sit at room temperature until completely cool and firm, about 2 hours.
Invert nougat onto a cutting surface and remove the parchment. The slab should be sticky side up. (The side that was exposed to air while setting should be dry and not sticky when touched.) Cut into 1-inch squares and place squares sticky side up on a parchment paper-lined baking sheet.