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Place turbinado sugar in a medium heavy-bottomed saucepan. Add 1/4 cup water and stir to combine. Place over high heat and cook, until sugar is dissolved and mixture becomes a very dark caramel color, about 10 minutes.
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Slowly whisk in milk, taking care as caramel may bubble and splatter. Place over medium-high heat and cook, whisking, until mixture is well combined. Remove from heat; set aside.
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In a large bowl, whisk together egg yolks and granulated sugar until well combined. Add half of the caramel mixture to the egg yolk mixture; whisk to combine. Return egg mixture to saucepan and whisk to combine. Place saucepan over medium-high heat and cook, whisking occasionally, until mixture is thickened, about 10 minutes.
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Meanwhile, set a medium bowl in an ice-water bath; set a fine-mesh sieve over bowl. When ice cream mixture has thickened, strain and let cool completely.
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When mixture is cooled, transfer to an ice cream maker and freeze according to manufacturer's instructions. Transfer ice cream to an airtight container and keep frozen up to 1 week.