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Preheat the oven to 350°F (180°C, or gas mark 4). Spread the walnuts on a parchment- or silicone-lined rimmed baking sheet.
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Bake for 10 minutes, or until the walnuts are toasted and fragrant, stirring every 5 minutes.
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Check the walnuts every few minutes to prevent burning.
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Cool to room temperature and chop. Store in a covered container until ready to use.
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Fill a large bowl with ice water.
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In a small bowl, combine 2 tablespoons (30ml) of the milk with the cornstarch, whisk and set aside.
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Combine the remaining milk with the heavy cream, sugar and honey in a medium saucepan and place over medium heat.
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Bring the milk mixture to a low boil.
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Cook until the sugar and honey dissolve, 3 minutes.
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Remove the milk mixture from the heat and gradually whisk in the cornstarch mixture.
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Return to a boil and cook over moderately high heat until the mixture is slightly thickened, about 1 minute.
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Pour into a medium bowl. Whisk in the rosemary and salt. Set the bowl in the ice water bath to cool, 20 minutes, whisking occasionally.
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Cover and refrigerate until well chilled, at least 4 hours or overnight.
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Once chilled, pour the ice cream base through a fine-mesh sieve to remove the rosemary pieces.
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Pour into an ice cream maker and churn according to the manufacturer’s instructions.
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When churning is complete, gently fold in the walnut pieces.
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Transfer to a freezer-safe container and freeze until firm, at least 4 hours.