Make the broth and strain it. Pluck the chicken pieces out and set them aside until they are cool enough to handle.
When the chicken is cool, pull off the skin and discard, then shred the meat coarsely, getting rid of any fat, gristle, etc. as you go. Set the chicken meat aside. The broth may be made and the chicken shredded up to 2 days before preparing the soup.
Heat a large pot of salted water to a boil.
Bring the strained broth, celery, and carrots to a boil in a large pot on medium heat. Taste the broth and add salt and pepper, if necessary. Adjust the heat so the broth is simmering and cook until the vegetables are tender but not mushy, about 10 minutes.
Meanwhile, stir the fideos into the boiling water, crushing each handful to break the noodles up as you go. Cook until tender but with a little bit of a bite, about 4 minutes. Drain the noodles and put them in a bowl. Add the olive oil and toss well.
When the vegetables are tender, stir the shredded chicken, avocado, cilantro and lime juice into the soup. Bring the pot of soup to the table, along with the noodles and lime wedges. Spoon some of the noodles into each bowl and ladle some of the soup over them. (Stir the soup with a ladle so you're sure to get some of each ingredient into each bowl.) Pass the lime wedges separately.
Homemade Chicken Broth:
Put the chicken, onions, and red pepper in a large stockpot. (Mine is 16 quarts.) Rinse the garlic heads in cold water, then peel off the papery skin. Break the garlic head into individual cloves. Rinse and cut the roots off the cilantro. Cut a piece of cheesecloth the size of a kitchen towel. Put the garlic cloves, peppercorns, bay leaves, achiote seeds and cilantro on the cheesecloth, bring the ends together to make a little pouch, and tie the loosely with kitchen twine. Give the pouch a couple of good whacks with your kitchen mallet, and add it to the pot.
Add enough cold water to cover the chicken by at least 2 to 3 inches. Drop in the ham hock, if using. Bring to a boil over medium-high heat, skimming foam off of the top as it forms. When the water comes to a boil, adjust the heat so the broth cooks at a steady, but not rolling boil. Cook for 30 minutes.
Check to make sure the breasts are cooked through, then remove them from the pan. Cook 15 minutes more, then remove the legs and thighs. Set the chicken aside to make another chicken dish. (If you're using backs, necks, wings, etc., you don't have to remove anything.) Leave the backs, necks and wings in to further flavor the broth.
Cook for 1 hour. Set a strainer over a large bowl and line the strainer with a couple of layers of cheesecloth. Carefully strain the broth through the cheesecloth. (Ladling it is the easiest and safest way.) Add salt, to taste, and let cool before refrigerating.
Yield: 3 to 3 1/2 quarts
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