Inspired by the classic French dish, large chunks of eggplant stand in for beef chuck in this vegan iteration. Eggplant is browned in olive oil, then braised in a sauce of red wine, aromatics and both fresh and dried mushrooms. Use any kind of fresh and dried mushrooms that you can find and like; they all work well here. And search for smaller, narrower Japanese, Chinese or Fairy Tale eggplants. Their size, when cut into disks, works perfectly for this dish, and their flesh cooks up more tender. But if all you can find is the common Italian purple eggplants, they will work well too; cut them into 1 1/2-inch pieces so they fit in a single layer in the pot.
Pour 2 tablespoons olive oil in the bottom of a large Dutch oven or enameled cast-iron pot. Add the tarragon sprigs, then turn the heat to medium-high. Once the leaves begin sizzling, continue to cook until they just stop sizzling and are translucent, 1 to 1½ minutes. Using tongs, remove the tarragon sprigs and leaves and place on a paper towel to drain. Reduce the heat to medium, and add another 2 tablespoons olive oil.
Add the fresh mushrooms, season with salt and pepper and cook, stirring occasionally, until the mushrooms give up their liquid and become golden brown, 6 to 8 minutes. Meanwhile, pour the vegetable stock into a small saucepan and bring to a boil. Turn off the heat and add the dried mushrooms; let stand until needed.
Using a slotted spoon, remove the fresh mushrooms from the pot and place on a plate. Add another ¼ cup olive oil to the pot. Season both sides of the eggplant pieces with salt and pepper, then place in a single layer in the pot (they will all fit). Cook, undisturbed, until golden brown on the bottom, 2½ to 3 minutes. Flip the eggplant pieces, pour in another 2 tablespoons olive oil around the eggplant, and cook until golden brown on the other side, 2 to 3 minutes more. The eggplant will not be cooked through, just browned on the cut sides. Transfer the eggplant pieces to a plate. Reduce the heat to medium-low.
Add the remaining 2 tablespoons olive oil to the pot, then add the onions, carrots and celery. Cook, stirring occasionally, until the aromatics are soft and deep golden brown all over, about 14 to 15 minutes. Meanwhile, heat the oven to 350 degrees.
Add the garlic and tomato paste to the vegetables and cook, stirring constantly, to cook out the raw taste of the tomato paste, about 2 minutes. Pour in the red wine and scrape the bottom of the pot to pick up all the browned bits on the bottom. Let the wine cook until reduced by half, about 2 minutes.
While the wine simmers, fish out the reconstituted dried mushrooms and finely chop them. Add them to the pot along with the vegetable stock, avoiding pouring in any sediment that collects at the bottom of the pan; discard the sediment. Add the thyme and bay leaves, then return the eggplant pieces to the pot in a single layer. Scatter the cooked fresh mushrooms over the eggplant and bring the liquid to a boil. Cover the pot and transfer to the oven. Cook until the eggplant pieces are very tender when pierced in the center with a paring knife but not falling apart, about 1 hour.
Transfer the pot to the stove. Uncover and, using tongs or a spoon, transfer the eggplant pieces to a serving platter. Remove and discard the bay leaves and thyme stems. Turn the heat to medium to bring the sauce in the pot to a simmer. Meanwhile, in a small bowl, stir together the remaining 2 teaspoons olive oil with the flour until it forms a thick slurry.
Scrape the slurry into the simmering sauce and cook, stirring, until the sauce thickens, about 1 minute. Stir in the vinegar, then taste the sauce and season with more salt and pepper. Spoon the sauce and mushrooms over the eggplant pieces. Break off the tarragon leaves and scatter them over the eggplant just before serving with couscous.
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