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1
Place the sliced garlic in a small saucepan with enough milk to cover. Bring to a gentle simmer, drain, and repeat twice more — three changes of milk in total, each at about 15 minutes. Drain the garlic after the final change and pat thoroughly dry. The cloves should be very tender and smell sweet rather than sharp.
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2
Combine the olive oil and butter in a heavy terracotta or enamelled vessel over the absolute minimum heat possible. Add the pre-treated garlic. Cook, stirring occasionally, for 20 minutes. The garlic must not colour at any point — at the first trace of gold the dish is already compromised. It is ready when the slices have collapsed to a soft, pale, near-invisible paste.
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3
Add the desalted, dried anchovies to the garlic oil. This is where the dish lives or dies: maintain the lowest possible heat and stir continuously for 10 to 15 minutes, pressing and working the fillets against the base of the vessel until every trace of fish structure is gone and the sauce is a single, loose, pourable, homogeneous emulsion — no anchovy fragments, no separated oil.
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4
Transfer the bagna cauda immediately to individual terracotta tajine vessels. Set each over a lit tea light. Bring to the table at once with the prepared vegetables alongside. The sauce must remain warm throughout the meal; if it cools, the fat congeals and the garlic turns sharp.