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.

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.

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.

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.

Stir the shredded chicken into the soup. Serve in soup bowls, garnished with a generous grating of cheese and additional olive oil, if desired.