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1
Pour the olive oil into a wide, heavy pot set over the lowest possible heat. Add the sliced onions and a pinch of salt. Cook, stirring occasionally, for 30 to 40 minutes. The onions are done when they are completely collapsed, deeply golden, and sweet-smelling — no rawness, no sharp bite, no pale colour at the centre of any strand.
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2
Add the tomatoes to the softened onion. Raise the heat to moderate and cook until the tomatoes break down and the mixture tightens into a cohesive, oil-glazed mass, about 10 minutes. For the winter variant, add the soaked, drained porcini here and use their strained soaking liquid as part of the water measure.
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3
Pour in 1 litre of cold water. Season with salt and a generous amount of black pepper. Bring to a simmer and cook uncovered for 20 minutes. The liquid should reduce to a concentrated, amber-coloured broth that carries the sweetness of the onion — not a thin, watery liquid.
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4
Lay a slice of stale pane sciocco in each warmed bowl. The bread must be stale so it absorbs the soup cleanly without dissolving into paste.
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5
Reduce the heat so the soup barely trembles. Crack the eggs one at a time directly into the simmering pot. Cover and leave undisturbed for 3 minutes. The egg is ready when the white is just set and opaque throughout but the yolk yields visibly to the lightest pressure — still liquid at the centre.
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6
Ladle the soup over the bread in each bowl, then lift one poached egg on top of each serving with a slotted spoon. Finish with a thread of raw extra-virgin olive oil and a spoonful of grated Pecorino. Bring the oil and Pecorino to the table so each diner may add more.