1. Caramel: Lightly brush bottom and sides of a 9-by-13-inch rimmed baking sheet with oil. Line with parchment, leaving a 2-inch overhang on long sides; lightly brush parchment with oil.

2. Bring cream, granulated sugar, butter, and corn syrup to a boil in a medium saucepan, stirring until sugar is dissolved. Reduce heat to medium-high; continue cooking, without stirring but swirling pan occasionally, until a candy thermometer registers 240 degrees, 8 to 10 minutes. Immediately remove from heat; stir in salt and vanilla.

3. Pour caramel onto prepared baking sheet; spread with an offset spatula. Let stand, uncovered, at least 30 minutes and up to 1 day.

4. Nougat: Sift together milk powder and confectioners' sugar into a bowl.

5. Place 1 1/4 cups granulated sugar, corn syrup, and 1/4 cup water in a small saucepan. Bring to a boil, stirring occasionally, until sugar is dissolved. Continue cooking, without stirring but swirling pan occasionally, until a candy thermometer registers 250 degrees, 8 to 10 minutes.

6. Meanwhile, as sugar starts to boil, beat together egg white and remaining 2 tablespoons sugar on medium speed until stiff but not dry peaks form. Reduce speed to low; as soon as sugar syrup reaches 250 degrees, carefully pour down side of bowl. Increase speed to high and continue beating until stiff peaks form, about 2 minutes more. Beat in vanilla, then chocolate. Fold in milk-powder mixture. (Nougat will appear grainy.)

7. With a rubber spatula, scrape nougat onto caramel; smooth with an offset spatula. Let stand, uncovered, until completely cool and crystallized, 8 to 12 hours.

8. Using parchment, lift candy slab onto a cutting board, keeping parchment underneath. With an oiled knife, cut into 9-by-1/2-inch strips; twist into spiral ropes. To serve, cut strips into 2-inch pieces, if desired. Candies can be wrapped individually in waxed paper and stored in an airtight container up to 1 week.