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1
Pour the olive oil into a wide, heavy pot over a low flame. Add the sliced onions, celery, and peperoncino flakes if using. Cook uncovered for 40 minutes, stirring occasionally, until the onion has completely dissolved into the oil and no individual onion piece retains its identity — undivided, cohesive sweetness is the mark of completion.
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2
Add the well-salted water and the strip of dried porcini if using. Raise the heat to bring the soup to a gentle simmer, then reduce and cook for a further 10 minutes. Taste: the broth must carry depth and seasoning — a water-based soup that is not aggressively seasoned tastes of nothing.
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3
Place one thick slice of stale bread into each warmed soup bowl. The bread must be sturdy enough to absorb the broth and swell without dissolving — it should hold its shape when pressed.
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4
Bring the soup back to a calm simmer. Crack each egg directly into the broth and poach for 2 minutes. The white must be fully set around the yolk; the yolk must yield and run when pressed with a spoon.
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5
Ladle the hot broth over the bread in each bowl, then transfer one poached egg on top of each slice. Grate Pecorino Romano generously over each portion and serve immediately — the cheese should begin to dissolve partially into the hot soup on contact.