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1
Preheat oven to 300 F/150 C. Pat the roast dry and sprinkle all sides with salt & pepper. Spread 2 Tbs. of flour out on a plate and lightly roll the roast in it and pat it to coat all sides.
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2
In a Dutch oven or large oven-proof pot, heat the olive oil over medium-high heat, and when oil is hot, add the roast. Let it brown on all sides, about 4 minutes per side, then remove and set aside in a bowl. Turn the heat to low and add the vinegar, followed quickly by the wine, and use a wooden spoon to scrape off any browned bits on the bottom and sides of pot. Return the roast to the pot and add the stock and enough water to nearly cover the roast (a half-inch or so peeking above the liquid is fine). Add the bay leaves and herbs, bring to a simmer, cover, and transfer the whole pot to the preheated oven.
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3
Cook, covered, for about 2 hours, turning the roast once or twice and checking doneness. When the roast flakes apart at the touch of the wooden spoon (by 2 1/2 hours of cooking at the longest), add the carrots, daikon & onion and the demi-glace if using, stir well, cover and return to oven. 15 minutes later, add the whole mushrooms, cover and return to oven. Cook another 15 to 30 minutes, until carrot and daikon are to desired softness and mushrooms are cooked through. Transfer the pot to the stovetop.
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4
Carefully remove the roast to a cutting board and tent with foil to keep warm. Taste the sauce and add more salt & pepper and a small splash of red wine vinegar to taste. Remove the bay leaves and thyme bundle. In a mug or small bowl, mix the softened butter with the remaining 2 Tbs. flour until a smooth paste, add some of the warm liquid broth, and stir in the mug until any lumps are gone. Add the mixture to the pot and heat over medium heat (uncovered) until it bubbles and thickens as you stir. Taste again and adjust seasonings.
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5
Cut the roast in thick slices against the grain and serve in bowls with the thickened sauce, vegetables, and a sprinkling of fresh parsley and lemon zest.
