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1
Combine the saffron threads with the white wine and leave to bloom for a full 10 minutes. The wine will turn a deep amber-gold: this colour and its depth of extraction defines the character of the finished sauce.
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2
Pat the monkfish medallions dry. Dust each medallion evenly in plain flour and shake off the excess. Warm the olive oil in a wide, heavy pan over a moderately high heat. Brown the medallions on all sides until a caramelised, golden crust forms — this exterior both thickens the sauce during the braise and gives the flesh structural integrity. Remove the medallions and set aside.
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3
In the same pan over moderate heat, sweat the diced onion and garlic in the residual oil until soft and translucent, without colouring. Add the tomato concassé and cook until it breaks down into a cohesive base, about 5 minutes. Pour in the saffron-steeped wine, add the bay leaves and thyme, and stir to combine.
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4
Return the browned medallions to the pan, nestling them into the braising base. Cover and simmer over a low heat for 20–25 minutes. The dish is ready when the flesh pulls cleanly from the central cartilage under gentle pressure from a spoon and yields with slight resistance — not rubbery, not collapsing.
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5
Remove the pan entirely from the heat. This is where the dish lives or dies: stir the room-temperature aioli into the sauce with deliberate, folding strokes until the braising liquid shifts from a thin orange-red to a pale, slightly opaque, golden emulsion that coats the medallions. Any residual heat from the burner after this point will break the emulsion, separating the oil and rendering the sauce greasy and irreparable. Serve immediately.