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1
Combine honey, water, and cream of tartar in a small sauce pan. Cover and heat over medium until boiling, swirling occasionally. Remove the cover and increase heat to medium-high. Swirl the pan frequently until the honey becomes a dark amber color. It with foam when you swirl it, and the foam will be pale at first. As the honey caramelizes the foam will become amber colored in places and eventually all over. Let the honey get fairly dark so it gets just a hint of bitterness. It should reach 250-260u00b0 F on a candy thermometer if you're testing.
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2
While the honey is boiling, combine the cream, salt, and butter in a glass measure or small saute pan. Warm in the microwave or on the stove top. Exact temperature isn't critical, but it should be too warm to bathe it.
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3
When the honey has caramelized, remove it from the heat and slowly pour in the cream mixture. It may froth up a bit, but not as dramatically as when making regular caramel syrup because the honey browns at a lower temperature. Whisk thoroughly, and transfer to a glass jar or measure. Allow to cool to room temperature, then cover and refrigerate until chilled.
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4
Ice cream prep: Whisk cornstarch and about 2 tablespoons of milk together in a small bowl and set aside. Combine the cream cheese and 3/4 teaspoon salt in a medium bowl, and whisk until smooth. Fill a large bowl with ice and water -- lots of ice!
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5
Combine the remaining milk, cream, sugar, honey and corn syrup in a 4- to 6-quart pan. Heat over medium-high, and when the mixture reaches a rolling boil set a timer for 4 minutes. After 4 minutes remove the milk mixture from the heat. Re-whisk the milk-cornstarch mixture you made earlier, then slowly whisk it into heated milk mixture. Return pot it to medium-high heat. Cook, stirring continuously, until the mixture thickens slightly, about 3 to 4 minutes. The mixture should form a thin film on the back of a wooden spoon when it is ready.
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6
Add about 1/2 cup of the milk mixture to the cream cheese and whisk vigorously until only small flecks of cream cheese are visible. Add another 1/2 to 1 cup of milk, and whisk until no flecks of cream cheese are visible. Whisk in the remaining milk mixture.
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7
Transfer the ice cream to a metal bowl that will safely fit into your ice bath bowl. Whisk occasionally while cooling, and allow to chill thoroughly (45u00b0 F is good). This will take at least 40 minutes, and you may need to add more ice to your bath. Taste for salt once chilled. You are looking for it to be obviously salty, but not painfully salty, and depending on what sea salt you have you may need to whisk in more, a little at a time. Make sure to allow it to dissolve well and sip some water before re-tasting.
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8
Freeze the ice cream base as directed by your ice cream machine's instruction manual. Transfer the ice cream into a storage container, drizzling with thick ribbons of sauce as you go, and ending with a drizzle of sauce. Place a piece of parchment on the surface of the ice cream, and place the container in the freezer. Freeze in the coldest part of your freezer until firm, at least 4 hours, before serving. (NOTE: the caramel stays nicely gooey in this ice cream, so I'd recommend serving in a dish rather than a cone).
