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1
Drain the soaked beans and place in a large saucepan. Cover generously with fresh cold water — no salt, no acid. Bring to a boil, skim any foam, then reduce to a steady simmer. Cook until the beans are completely tender throughout, 45–75 minutes depending on age. They must yield without resistance at the centre. Drain and set aside.
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2
Warm the olive oil in a wide, heavy pan over a medium-low flame. Add the onions and sweat, stirring occasionally, until fully soft and translucent without colour — 10 to 12 minutes. The onions must offer no resistance and carry no raw sharpness.
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3
Add the carrot and celery to the softened onions. Stir to coat in the oil and continue to cook gently for 5 minutes, until the vegetables begin to soften at their edges.
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4
Stir in the tomato paste and cook for 2 minutes, then add the fresh tomato and bay leaves. Cook together until the tomato breaks down into the oil and the raw smell dissipates — the fat should begin to separate slightly at the surface.
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5
Add the drained cooked beans. Pour in just enough water to produce a slightly saucy consistency — the beans should be bathed in liquid but not swimming. This is where the dish lives or dies: too much water and the beans will never absorb the olive oil and vegetable flavours; too little and the mixture scorches before the absorption is complete. Adjust accordingly before the heat rises.
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6
Simmer very gently, uncovered, for 20 to 30 minutes, until the beans have absorbed the olive oil-tomato cooking liquid and the sauce has thickened to a coating consistency around the beans. Stir carefully once or twice. The surface should show a gloss of olive oil.
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7
Remove from the heat. Squeeze over the lemon juice, season with salt, and stir once. Taste: the acidity should cut the richness of the olive oil without dominating. Remove the bay leaves. Allow the pilaki to cool completely — to cold or room temperature — before serving. The flavour deepens as it cools.