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1
Bring sugar and 1/2 cup water to a boil in a small saucepan over medium-high heat, swirling occasionally to dissolve sugar. As soon as the sugar is dissolved, add walnuts and bring syrup back up to a simmer. Cook, swirling saucepan occasionally for even cooking, until walnuts look slightly translucent and are softened, 30-35 minutes.
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2
Drain walnuts in a mesh sieve and arrange in a single layer on a rimmed baking sheet lined with a silicone baking mat or parchment paper. Sprinkle walnuts with salt.
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3
Place a rack in upper third of oven and preheat to 300u00b0F. Bake walnuts in oven, stirring once or twice, until well browned, 15-18 minutes. Let cool, then break up into small pieces.
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4
Place chocolate in a medium bowl; set aside. Bring cream and milk to a bare simmer in a large saucepan and remove from heat.
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5
Vigorously whisk egg yolks and sugar in another medium bowl until pale yellow and sugar is dissolved (rub between fingers to check), about 2 minutes.
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6
Gradually pour cream mixture into egg yolk mixture, whisking vigorously. Immediately pour egg yolk mixture into saucepan and cook over medium heat, stirring constantly with a heatproof rubber spatula and making sure to get into the corners of the pan, until an instant-read thermometer registers 175u00b0F (custard should be thick enough to coat a spoon; do not let it simmer or allow thermometer to go above 180u00b0F).
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7
Pour custard over reserved chocolate and let sit until chocolate is melted, about 5 minutes. Using an immersion blender or a whisk, blend until emulsified (if using a whisk, you may want to strain budino through a fine-mesh sieve after to ensure a silky texture). Add kosher salt and 1 Tbsp. oil to chocolate mixture and blend to incorporate.
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8
Divide budino evenly among six 8-12-oz. glasses. Cover with plastic wrap and chill until firm, at least 1 hour.
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9
Top each budino with candied walnuts, sprinkle with sea salt and pepper, and drizzle with a little oil. Serve with scoops of gelato if desired.
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10
Walnuts can be candied 1 week ahead. Store airtight at room temperature.
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11
Budinos (without toppings) can be made 3 days ahead. Keep chilled.
