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**Ice Cream:**
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1. Toast the walnuts: Spread the chopped walnuts on a baking sheet and toast in a preheated oven at 350°F (175°C) for 5-7 minutes, or until fragrant and lightly golden. Let cool.
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2. Make the custard base: In a saucepan, combine the Jersey milk and cream. Heat over medium heat until just simmering, stirring occasionally. Do not boil.
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3. Whisk egg yolks and sugar: In a separate bowl, whisk the egg yolks and caster sugar together until pale and thickened.
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4. Temper the egg yolks: Slowly drizzle a small amount of the hot milk mixture into the egg yolks, whisking constantly to prevent scrambling. Repeat until the egg yolk mixture is warmed through.
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5. Cook the custard: Pour the egg yolk mixture back into the saucepan with the remaining milk mixture. Cook over low heat, stirring constantly with a wooden spoon, until the custard thickens enough to coat the back of the spoon. Do not boil.
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6. Incorporate Black Butter: Remove from heat and stir in the Jersey Black Butter and vanilla extract until well combined.
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7. Chill: Pour the custard base through a fine-mesh sieve into a clean bowl. This will remove any lumps. Cover the bowl with plastic wrap, pressing the wrap directly onto the surface of the custard to prevent a skin from forming. Refrigerate for at least 4 hours, or preferably overnight.
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8. Churn the ice cream: Once the custard base is thoroughly chilled, churn it in an ice cream maker according to the manufacturer's instructions.
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9. Fold in walnuts: During the last few minutes of churning, fold in the toasted walnuts.
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10. Freeze: Transfer the ice cream to an airtight container and freeze for at least 2 hours, or until firm.
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**Spiced Pear Compote:**
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1. Sauté pears: In a saucepan, melt the Jersey butter over medium heat. Add the diced Jersey pears and cook for 5-7 minutes, or until slightly softened.
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2. Add spices and sugar: Stir in the brown sugar, cinnamon stick, star anise, and ground ginger. If using the potato, add this now. Cook for another 5 minutes, stirring occasionally, until the pears are tender and the sugar has dissolved.
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3. Simmer: Reduce the heat to low, add the lemon juice, and simmer for 10-15 minutes, or until the compote has thickened slightly. Remove the cinnamon stick and star anise.
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4. Cool: Let the compote cool slightly before serving.
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**Serve:** Scoop the Jersey Black Butter and Walnut Ice Cream into bowls and top with the spiced pear compote.