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1
Bring a large pot of well-salted water to a full boil. Meanwhile, if colour retention is a priority, blanch the basil leaves in boiling water for 5 seconds, then transfer immediately to ice water. Drain and pat dry. The leaves should be a vivid, unbroken green with no dark bruising.
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2
In a stone mortar, work the garlic cloves and coarse salt to a smooth paste using large circular grinding strokes of the pestle — not pounding. Add the pine nuts and grind to a rough paste. Add the basil leaves in two or three batches, grinding each batch into the preceding paste before adding the next. The mortar will smell intensely of cut grass and sweet garlic; the paste should be cohesive and pale green. Add the Parmigiano-Reggiano and Pecorino Sardo and work briefly to incorporate. Finally, pour the Ligurian olive oil in a thin stream, rotating the pestle to emulsify. This is where the dish lives or dies: the mortar's grinding action ruptures the basil cells without generating blade-friction heat, preserving the volatile aromatics and fresh green colour that a blender at speed destroys within seconds.
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3
Mix the flour with enough lukewarm water to form a firm, smooth dough. Pinch off a small piece — roughly the size of a grape — and roll it between both palms on a lightly floured surface, applying slight diagonal pressure, until a short twisted coil forms. Repeat with the remaining dough. The finished trofie should feel dry to the touch and hold their tight spiral without unravelling.
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4
Drop the diced potato and cut green beans into the boiling salted water 5 minutes before the pasta is to be added. After 5 minutes, add the trofie. Cook until the trofie are tender throughout and the potato yields cleanly when pressed — the cooking water will have taken on a faint starchy, savoury quality. Reserve a ladleful of this cooking water before draining.
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5
Place the pesto in a large warmed bowl. Loosen it with two or three tablespoons of the reserved starchy cooking water — never raw water — stirring until the sauce is fluid but still coating. Drain the trofie, potato, and green beans together and transfer immediately to the bowl. Toss to dress evenly. The pasta should glisten with a coating of bright green pesto, and the sauce should not pool at the bottom of the bowl.