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In a medium saucepan set over medium-high heat, bring the milk, 1 cup (227g) cream, and sugar just to a boil, stirring occasionally.
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In a medium bowl, whisk the eggs until frothy.
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Slowly add half the hot milk/cream mixture to the eggs, whisking constantly; then pour this mixture back into the saucepan.
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Cook the custard over medium-low heat, stirring constantly, until it thickens enough to coat the back of a spoon; it should never come to a boil. When thick enough, the custard should register 170°F on a digital thermometer.
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Pour the custard through a fine-mesh strainer into a large bowl (a metal bowl is best). Set it aside.
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Wash and dry the saucepan, and add the caramel, remaining 1/4 cup (57g) cream, salt, and vanilla.
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Warm the mixture over medium-low heat, stirring to help soften the caramel. Cook and stir until completely smooth.
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Stir the warm caramel into the warm custard completely, leaving no streaks.
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Cover the surface of the custard mixture with a piece of plastic wrap and refrigerate until chilled throughout, at least 4 to 6 hours or up to overnight.
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Freeze the custard in an ice cream maker according to the manufacturer’s instructions.
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Transfer the finished ice cream to an airtight container and place it in the freezer until ready to serve.