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1
Heat the oil in a heavy wide pan over medium-low heat. Add the sliced onions and cook, stirring at intervals, for 30 minutes or longer, until deeply golden-brown and collapsed. The onions are ready when they are uniformly mahogany in colour and smell of sweet caramel rather than raw allium.
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2
Add 100 ml of the red wine to the caramelised onions. Raise the heat to medium and cook, stirring, until the wine has fully evaporated and no sharp alcohol note remains. The pan should smell of cooked fruit and browned onion, with no vinous edge.
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3
Add the sliced mushrooms to the pan and sauté until they have released their liquid and the liquid has cooked away. The mushrooms are done when the pan bottom is dry and the mushrooms are lightly coloured.
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4
Add the canned demi-glace sauce, the remaining 100 ml red wine, the beef stock, and the tomato paste. Stir to combine. Bring to a gentle simmer and cook for 15 minutes, stirring occasionally, until the sauce coats the back of a spoon and the surface has a glossy, slow-bubbling appearance. This is where the dish lives or dies: the sauce must reach a coating consistency and deep meaty-tomato depth before the beef is introduced — a thin, flat-tasting sauce cannot be corrected once the beef goes in.
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5
Add the thinly sliced beef to the sauce. Stir gently to separate the slices and cook for no more than 5 minutes, until the beef is just cooked through and no longer pink. Prolonged cooking at this stage toughens the thin slices irreversibly.
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6
Remove the pan from the heat. Add the cold butter in pieces and stir until fully melted and incorporated. Rest the hayashi, covered, for at least 20 minutes off the heat. The sauce will deepen in colour and the flavours will integrate during the rest.
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7
Mound the cooked white rice on each plate, either in a dome or pressed in a bowl and turned out. Ladle the hayashi sauce generously alongside or over the rice. Serve immediately.