Make shortbread crust:
Preheat oven to 375°F with rack in middle. Butter a shallow 9- to 9 1/2-inch square baking pan (1 to 1 1/2 inches deep). Line bottom and 2 sides with parchment paper, leaving an overhang. Butter parchment.
Combine brown sugar, flour, and salt in a food processor and process until blended. Add butter and blend until it is incorporated. Add vanilla and up to 1 tablespoon ice water and process until a soft dough forms.
Spread dough evenly in baking pan using an offset spatula or back of a spoon, then prick all over with fork.
Bake until golden, about 20 minutes, then cool completely in pan on a rack, about 30 minutes.
Make chocolate dulce de leche:
Bring cream and dulce de leche to a simmer in a small heavy saucepan, stirring with a wooden spoon until dulce de leche has dissolved.
Whisk together yolks in a bowl, then slowly whisk in hot cream mixture. Return to pan and cook over medium heat, stirring constantly, until pan is visible in tracks of spoon and mixture registers 170°F on an instant-read thermometer. Remove from heat and whisk in salt and chocolate until melted.
Pour chocolate mixture over cooled shortbread and chill, uncovered, until cold and set, at least 2 hours.
Run a small knife around edges to loosen, then transfer to a cutting board using parchment. Cut with a hot clean knife (dip in hot water and wipe clean between cuts) into 64 bars. Chill until ready to serve.
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