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1
Pound the garlic with the coarse salt in a marble mortar until a smooth paste forms. Add the pine nuts and reduce them to a paste. Introduce the basil leaves in successive small batches, working the pestle in a rotary motion rather than crushing downward, until the leaves are fully incorporated and the paste turns a uniform, bright green. Work in the grated Parmigiano and Pecorino. Finally, pour in the olive oil in a thin stream, stirring with the pestle until the pesto is cohesive, glossy, and a vivid, unoxidised green — this is the colour that confirms the aromatics are intact.
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2
Bring a large pot of well-salted water to a rolling boil. Add the potato cubes and green beans together. Cook for 5 minutes — the beans should be tender but retain a slight resistance and hold their colour.
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3
Add the trofie to the same pot without draining the vegetables. Cook for a further 8–10 minutes until the trofie are cooked through and their twisted surfaces feel firm but yielding when pressed between two fingers. Reserve at least 200 ml of the cloudy cooking water before draining.
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4
Transfer the pesto to a large serving bowl. Drain the trofie, green beans, and potato together and add them immediately to the bowl while hot. Begin tossing, adding the reserved pasta water one tablespoon at a time. Continue until the pesto coats every surface in a flowing, evenly distributed sauce — not dry, not pooled. The starchy water loosens the pesto without diluting it; the dish is ready when no uncoated pasta or dry pesto remains visible.