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Put two coupe or martini glasses in the freezer. A frosty glass keeps the drink cold and the foam intact.
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Add the sugar and 2 tablespoons of water to a small, light-colored saucepan (light makes it easier to judge the color). Set it over medium heat and stir just until the sugar dissolves. Then stop stirring and let it bubble, swirling the pan gently now and then. After 5–7 minutes it will turn a deep amber, like a well-toasted marshmallow.
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Take the pan off the heat and pour in the ½ cup of hot water. It will spit and steam dramatically, so stand back and use a long spoon. Return the pan to low heat for a minute, stirring, until any hardened bits dissolve. Stir in the salt and vanilla.
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Pour the syrup into a jar and let it cool to room temperature. While you wait, brew your espresso and let it cool too. Hot coffee melts the ice and gives you a watery drink.
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Fill a cocktail shaker with ice. Add the vodka, coffee liqueur, espresso, 1½ oz of caramel syrup, and a tiny pinch of salt. Shake for a full 15–20 seconds. That energetic shake is what creates the foam.
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Rub a little syrup around half the rim of each glass and press it into flaky salt. Strain the cocktail into the glasses through a fine strainer if you have one. Top with three coffee beans and a few extra drops of syrup.