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1
Place the tripe strips in a large pot of cold water. Bring to a full boil, blanch for 5 minutes, then drain and discard the water entirely. Rinse the tripe under cold running water. The tripe should carry no residual odour.
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2
Warm the olive oil in a wide, heavy-bottomed pan over a medium flame. Add the onion, celery, and carrot. Cook gently, stirring occasionally, until the soffritto is fully softened and beginning to colour at the edges — the vegetables must be sweet and yielding, not raw.
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3
Add the crushed tomatoes to the soffritto. Cook over a moderate heat, stirring regularly, until the sauce has reduced to a dense, glossy mass that holds its shape on the spoon and no longer releases free liquid. This is where the dish lives or dies: a watery tomato base at this stage will produce a thin, coating-less sauce that no amount of subsequent cooking will correct.
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4
Add the blanched tripe strips to the dense tomato base. Add the Parmigiano rind. Stir to coat the tripe thoroughly with the sauce. Reduce the heat to low, cover, and braise patiently, stirring occasionally, until the tripe is completely tender and yielding — it must be soft enough to cut with a spoon. The sauce will continue to thicken and gloss around the tripe as it braises.
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5
In the final 10 minutes of cooking, add the fresh basil leaves and stir through. Remove and discard the Parmigiano rind. The basil should be just wilted and fragrant, not darkened or bitter.
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6
Taste and correct for salt. Serve immediately in warmed bowls. Pass the grated Parmigiano Reggiano at the table — each diner adds it generously over their own portion. The finished sauce must coat every strip of tripe in a thick, glossy film; it must not pool liquid at the base of the bowl.