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Place potatoes in large Dutch oven or soup pot.
Fill pot with water until potatoes are submerged and covered by 1 inch water. Add 1 teaspoon sea salt. Bring mixture to boil over medium-high heat and continue cooking 20-25 minutes, until potatoes are easily pierced by fork.
Meanwhile, in tall-sided saucepan, heat neutral oil over medium-low heat.
Trim roots from 2/3 of scallions and cut into 1-inch pieces. Pulse scallions in food processor until finely minced. Add pinch sea salt and pulse until wet and puréed. Carefully add purée to warm oil. (Oil should be warm, not hot, but be careful of splatters.)
Raise heat to high and cook, stirring, until boiling. Turn heat down to maintain mellow but insistent sizzle. Cook 35 minutes, stirring every few minutes to make sure bottom and sides aren’t sticking, until purée starts to turn brown in spots and goes from thick and clumpy to very liquid in pot.
While scallions cook, preheat oven to 425 degrees and drizzle 1 tablespoon olive oil over large, rimmed baking sheet. Brush oil so it’s evenly distributed over sheet.
When potatoes are done, drain in large colander and let cool 5 minutes. Once they can be handled safely, evenly distribute them over prepared baking sheet and use potato masher or serving fork to gently smash each potato to height of ½ inch. (Thinner potatoes are more crispy.)
Drizzle remaining 2 tablespoons olive oil over smashed potatoes. Sprinkle with sea salt and freshly cracked black pepper. Bake 25 to 30 minutes, until potatoes are golden on edges.
Meanwhile, trim roots off remaining scallions and cut into 1-inch pieces. Pulse in food processor until finely minced but not puréed. Stir minced scallions into cooked scallions, and turn heat up to high. Bring sauce back to boil, stirring, and then remove to heatproof bowl to cool to room temperature. When cool, season with more sea salt, to taste.
When potatoes are done, transfer to platter and drizzle with scallion sauce.