1. Stir together sugar, corn syrup, water, and salt in a heavy, 2-quart saucepan; attach a candy thermometer to side of pan. Place pan over medium-high, and cook, gently stirring occasionally with a wooden spoon, until sugar dissolves and mixture boils. Once sugar syrup is clear and thickened, cook, undisturbed, until thermometer reaches 255°F (hard-ball stage), about 6 to 8 minutes. Remove from heat.
2. Place egg whites in bowl of a stand mixer fitted with a whisk attachment. Beat egg whites on medium speed until foamy. Increase speed to high, and beat until egg whites are thick and stiff and able to hold a soft peak.
3. With mixer running on low speed, carefully pour hot syrup down side of bowl into beaten egg whites, pouring slowly and steadily. Use only the syrup that pours easily out of pan into bowl (no need to scrape pan). (Expect steam and the aroma of egg whites cooking in the syrup’s heat.)
4. Increase speed to high, and continue beating until syrup is well incorporated into egg whites. Stop just long enough to scrape down sides of bowl, and add vanilla. Continue beating on high speed until the mixture begins to lose its gloss and shine and is thick enough to hold its shape rather than pool into a puddle, 6 to 10 minutes.
5. Add 1⁄2 cup chopped peppermint candy, and beat on medium speed just until evenly combined. Remove bowl from mixer, and use 2 spoons to quickly scoop out small balls of candy (about 1 tablespoon each), placing them on prepared baking sheets; don’t allow candies to touch.
6. Garnish candies quickly with remaining 1⁄4 cup chopped peppermint. Let divinity cool, undisturbed, until set with a matte finish, 1 to 2 hours.
7. Carefully transfer candies to an airtight container or candy tin, placing parchment paper or wax paper between layers and handling candies gently.