To Toast the Walnuts: Preheat the oven to 350°F (180°C) and line a baking sheet with parchment paper. Toss the walnuts with a little olive oil and a pinch of salt, then spread them in a single layer on the prepared baking sheet, leaving some space between them. Bake for 5–7 minutes, or until lightly toasted and fragrant. Keep a close eye on them, as walnuts can burn quickly!
In a large saucepan over high heat, combine the potatoes, salt, and enough cold water to cover. Bring to a boil, then reduce the heat to maintain a gentle simmer. Cover and cook for about 25 minutes, or until the potatoes are fork-tender.
Using a food processor, blend the walnuts, basil leaves, mint leaves, onion, ground cumin, and jalapeño until finely chopped and combined into a green paste.
Drain the potatoes, reserving ½ cup of the cooking liquid. Mash them with a potato masher, or, alternatively, pass them through a food mill set over a large bowl. Turn the crank to press the potatoes through the mill until smooth, scraping any remaining potato from the bottom of the mill as needed.
Add the green herb paste, olive oil, and reserved ½ cup of cooking liquid to the mashed potatoes. Stir until well combined and smooth.
Mix well, taste, and adjust the seasoning with salt as needed.
Spread the mashed potatoes in a shallow serving bowl and smooth the surface with the back of a spoon. Drizzle with olive oil and garnish with walnut halves. Absolutely sublime!