Candied Walnuts:
1/2 cup sugar
1 cup coarsely chopped walnuts
Kosher salt
Budino and assembly:
1 cup bittersweet chocolate wafers (disks, pistoles, fèves; preferably 70%–74% cacao) or 5 oz. chocolate, finely chopped
1 cup heavy cream
1 cup whole milk
6 large egg yolks
1/4 cup sugar
3/4 tsp. kosher salt
1 Tbsp. extra-virgin olive oil (preferably Olio Verde), plus more for serving
Flaky sea salt
Freshly ground black pepper
Salted caramel gelato or ice cream (for serving; optional)
Candied Walnuts:
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.
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.
Place a rack in upper third of oven and preheat to 300°F. Bake walnuts in oven, stirring once or twice, until well browned, 15–18 minutes. Let cool, then break up into small pieces.
Budino and assembly:
Place chocolate in a medium bowl; set aside. Bring cream and milk to a bare simmer in a large saucepan and remove from heat.
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.
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 175°F (custard should be thick enough to coat a spoon; do not let it simmer or allow thermometer to go above 180°F).
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.
Divide budino evenly among six 8–12 oz. glasses. Cover with plastic wrap and chill until firm, at least 1 hour.
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.