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In a large deep pot, heat the olive oil over high heat. Add the onion and carrot and cook, stirring, until soft and translucent, 8 to 10 minutes. Stir in the garlic.
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Add the chicken, tomatoes with their juices, bay leaf, thyme, oregano, and allspice to the pot. Add water to cover the chicken by 3 inches (7. 5 cm). Bring to a boil, season generously with salt and pepper, reduce the heat to low, and simmer until the chicken is falling off the bone, about 1 1/2 hours.
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Using kitchen tongs or a slotted spoon, carefully remove the chicken from the pot and transfer it to a bowl. Let cool. When cool enough to handle, remove the meat from the bones, discarding the bones, and shred or chop the meat. Discard the skin, if desired.
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Remove the herb sprigs, bay leaf, and allspice from the pot. Add enough water to the pot to make 3 quarts (3 L) of liquid total. Stir in the tomato paste and honey. Add the noodles and bring the liquid to a boil. Cook according to the package directions.
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Stir the shredded chicken into the soup. Serve in soup bowls, garnished with a generous grating of cheese and additional olive oil, if desired.