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To make Caramel Sauce:.
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Bring cream to just under a boil over medium heat in a medium saucepan.
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Combine sugar, lemon juice, and 3 tablespoons water into a separate saucepan. (Be sure to use a heavy pan, such as a saucier) Place over medium heat, and, without stirring, bring to a simmer. (If sugar crystals form, dissolve with a wet pastry brush) Continue to cook while swirling the pan around until it becomes a deep caramel color, about 6 to 8 minutes (depending on the weight of the saucepan) let the color be your guide.
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Slowly add hot cream, stirring constantly with a wooden spoon. (Caution, mixture will spatter.).
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Cook 1 minute to dissolve caramel. Remove from heat.
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Stir in vanilla, kosher salt, and rum (if desired).
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Let cool. Store refrigerated in an airtight container up to 5 days; bring to room temperature before using.
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To make milkshake:.
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Process 1/2 cup caramel sauce and all ingredients in a blender until smooth. Serve in individual glasses, and top with additional caramel sauce, if desired.