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1
Place the fish carcasses and heads in a large saucepan with the cold water, onion, celery, bay leaves, thyme, rosemary, and fennel frond. Bring to a gentle simmer — not a rolling boil — and cook for thirty minutes, skimming any foam that rises. The broth is ready when it is pale gold, lightly fragrant with maquis aromatics, and tastes clearly of fish. Season fully with fine sea salt at this stage; the aioli will not rescue an under-seasoned broth. Strain through a fine-mesh sieve, pressing the solids lightly; discard the solids and return the clear broth to a clean pan.
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2
While the broth simmers, prepare the aioli. Whisk the crushed garlic paste and a pinch of salt into the egg yolks in a bowl until pale. Add the Corsican olive oil in the thinnest possible stream, whisking constantly, until the emulsion is thick, glossy, and holds a ribbon. Add the lemon juice, whisk briefly to incorporate. Set aside uncovered at room temperature — the aioli must not be cold when added to the broth.
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3
Bring the strained, seasoned broth to a gentle simmer (approximately 90°C). Lower the fish fillets into the broth, reduce the heat immediately, and poach for eight minutes. The fish is done when the flesh yields to light pressure and separates cleanly along the grain; it must not flake apart in the broth. Lift the fillets carefully from the broth and keep warm. Reserve the poaching broth.
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4
This is where the dish lives or dies. Reduce the heat under the poaching broth until it is below a simmer — no higher than 85°C; an active simmer will break the emulsion beyond recovery. Begin adding the room-temperature aioli in a thin, steady stream, whisking constantly in one direction. Continue until all the aioli is incorporated and the broth has transformed into a cream-coloured, velvety soup that coats the back of a spoon. Do not allow the soup to boil at any point after this stage. If the surface begins to show an oily sheen, remove the pan from the heat immediately and whisk vigorously while adding a small measure of room-temperature water or a spoonful of fresh aioli to arrest the break.
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5
Taste the finished soup for seasoning. Ladle into warmed wide bowls. Lay the poached fish on top of the broth. Serve immediately with the garlic-rubbed pain de châtaigne on the side. The soup is ready to serve when it is uniformly cream-white, velvety, and holds its emulsion with no oil separating at the surface.