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1
Render the lardons in a wide, heavy pan over medium heat until golden and crisp. Remove with a slotted spoon and set aside. In the rendered fat, sauté the button mushrooms until coloured, then remove. Glaze the pearl onions separately in a little butter until tender and set aside with the lardons and mushrooms. The garnish is ready when each element is cooked through and holds its shape.
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2
Return the pan to medium heat. Sweat the shallots and minced garlic in the remaining fat until softened and translucent. Add the flour and cook briefly, stirring, until the raw smell is gone — one to two minutes.
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3
Pour in the full 750 ml bottle of Burgundy and reduce by two-thirds over steady heat. This concentration is the essential act of the dish: the wine must lose its sharp, raw edge and thicken perceptibly. The surface will deepen in colour and the bubbles will slow. This is where the dish lives or dies — under-reduction leaves the sauce thin and acidic; no subsequent step can correct it.
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4
Add the demi-glace or rich brown stock and the bouquet garni. Simmer for 20 minutes until the sauce coats the back of a spoon with a deep, wine-dark gloss. Drag a finger across the spoon: the line must hold cleanly.
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5
Strain the sauce through a fine sieve, pressing lightly, then return it to a clean pan. Swirl in the 30 g of cold butter, piece by piece, over low heat without boiling. The sauce should be glossy and slightly viscous. Return the lardons, mushrooms, and glazed onions to the pan and keep warm.
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6
Bring a shallow, wide pan of water to a bare simmer — not a rolling boil — with the tablespoon of white wine vinegar. Working one at a time, slide each egg from its cup into the water. Poach for 3 to 3.5 minutes until the whites are fully set and opaque but the yolks yield entirely to gentle pressure. Lift with a slotted spoon, drain on a clean cloth, and trim any ragged edges of white.
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7
Place a garlic-rubbed croûton on each warm plate. Set a poached egg on the croûton. Spoon the meurette sauce generously over and around, distributing the lardon, mushroom, and onion garnish evenly. The dish is complete when the diner breaks the yolk and it mingles with the wine sauce in a rich, golden-purple stream.