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1
Bring onion, carrot, smashed garlic, beans, and 14 cups water to a boil in a medium pot, reduce heat, and simmer, stirring occasionally, until beans are creamy but still intact, 70-90 minutes (time will vary depending on size and age of beans). Using a slotted spoon, transfer beans to a medium bowl; discard onion, carrot, and garlic.
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2
Return broth in pot to a boil and add farro. Reduce heat and simmer, stirring occasionally, until grains are tender, 25-30 minutes. Remove from heat and return beans to pot; season generously with salt. Let sit 15 minutes to infuse beans and grains with flavor. (If you have the time, cook the beans and the grains a day ahead so they can really soak up the broth. If not, no worries-just keep going.)
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3
Cook chopped garlic, chiles, and 1/3 cup oil in a medium skillet over medium heat, stirring often, until garlic is golden, about 4 minutes. Remove from heat and mix in rosemary. Transfer chile-garlic oil to a small bowl; season with salt.
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4
Wipe out skillet and heat remaining 3 Tbsp. oil over medium-high. Cook mushrooms, undisturbed, until browned underneath, about 2 minutes. Toss and continue to cook, tossing every so often, until mushrooms are browned and crisp in most spots, another 6-8 minutes. Splash vinegar over mushrooms, season with salt and pepper, and give skillet a few shakes to combine.
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5
Ladle stew into bowls and top each with some crisped mushrooms and an egg yolk (if using); drizzle with chile-garlic oil.
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6
Break egg yolk and stir into soup just before eating to give it a more silky consistency.
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7
Stew (without mushrooms, egg yolks, and chile-garlic oil) can be made 3 days ahead. Let cool; cover and chill.
