Heat 2 tablespoons olive oil in a large Dutch oven over high heat. Working in batches so they brown rather than steam, sear the mushrooms 4-5 minutes per batch until deeply golden. Remove and set aside.
Reduce heat to medium and heat 1 tablespoon olive oil in the same pot. Add onion and carrots and cook 6-7 minutes until softened. Add garlic and tomato paste and cook 1 minute.
Pour in the red wine, scraping up any browned bits, and simmer 3-4 minutes to reduce slightly. Add the broth, thyme, and bay leaf. Return the mushrooms to the pot, bring to a simmer, cover, and cook 25-30 minutes until deeply flavorful.
Stir the cornstarch slurry into the stew and simmer uncovered 3-4 minutes until glossy and slightly thickened. Discard the thyme sprigs and bay leaf, then stir in the remaining tablespoon of olive oil. Season with salt and pepper.
While the stew simmers, bring the water or broth to a boil in a saucepan. Whisk in the polenta in a steady stream, then reduce heat to low and cook, stirring often, 20-25 minutes until thick and creamy. Stir in the olive oil, nutritional yeast (if using), and salt.
Spoon the polenta into bowls, top generously with the mushroom bourguignon, and serve immediately.
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