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In a dutch oven, heat 1 Tablespoon of olive oil over high heat.
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Add mushrooms and pearl onions and sauté for 3-4 minutes until they are seared and taking on some color, but before the mushrooms start releasing liquid.
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Remove the mushrooms and onions to a bowl and set aside.
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Add the second tablespoon of olive oil to the dutch oven and then turn the heat down to medium.
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Add carrot, onion, thyme, salt, and pepper to the dutch oven and sauté for 5 minutes until the onion has softened and taken on some color.
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Add the garlic and sauté for an additional 1 minute.
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Taste and season with additional salt and pepper if needed.
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Add wine to the dutch oven and deglaze, scraping any stuck bits off of the bottom of the dutch oven with a wooden spoon.
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Turn the heat up and reduce the wine by half, approximately 5 minutes.
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Add the tomato paste and the stock.
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Add back the mushrooms and pearl onions, along with any juices that have collected in the bowl.
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Bring the mixture to a boil.
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Reduce to a simmer and simmer for 15 minutes, or until the pearl onions and mushrooms are tender.
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In a small bowl, combine the butter and flour with a fork until it forms a paste.
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Stir the paste into the simmering liquid.
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Simmer for an additional 10 minutes.
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If the sauce is still thin, turn up the heat and reduce until slightly thickened.
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Taste and season with salt and pepper as needed.
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Serve over egg noodles (or mashed potatoes, polenta or gnocchi as desired).
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Garnish with chopped Italian parsley.