Put 2 tablespoons oil in a large pot over medium-high heat. Pat the chicken dry with a paper towel.
Put the chicken, skin side down, fresh or dried chiles, cilantro sprigs, tomato paste and lime, cut into quarters, in the pot, sprinkle with salt, and cook, undisturbed, until browned, 5 or 6 minutes. Trim and peel the onion. Trim and peel the carrots. Trim the celery stalks, reserving any leaves. Cut the onion, carrots, and celery stalks into 1/2-inch chunks. Peel garlic cloves and smash them with the flat side of a knife blade.
Add the vegetables and garlic to the pot with the chicken. Add 8 cups water, stirring to scrape the chicken and any browned bits from the bottom of the pot. Add 5 bay leaves and a pinch each of salt and pepper.
When the broth comes to a boil, adjust the heat so that it bubbles steadily. Cook, undisturbed, until the meat begins to loosen from the bones and is cooked through, 15 to 20 minutes. Chop the celery leaves.
When the chicken is beginning to fall off the bone, bring the broth to a rolling boil and add 8 ounces egg noodles. Start tasting after 5 minutes; when the noodles are tender but not mushy, turn off the heat.
Fish out the bay leaves and lime quarters. You can also remove the chilies and herbs if you prefer. Taste and adjust the seasoning, divide the soup among 4 bowls, garnish with celery leaves, and serve.