In a Dutch Oven or large stock pot, add chicken, onion, garlic, and 1 tablespoon salt (add the rest of your broth vegetables if making full-blown chicken broth; see notes). Add cold water to cover and bring to boil. Reduce heat, cover, and simmer until chicken is cooked through, about 20 to 30 minutes.
Remove chicken from pot. Shred chicken, discarding skin and bones (you should have about 4 cups chicken). Remove and set aside 1 cup cooking liquid (chicken broth). See notes for how to save the remaining chicken broth.
To make the sauce:
Meanwhile, in a medium saucepan bring 4 cups water to boil. Add tomatoes and boil until cooked through, about 5 minutes. Remove and transfer to blender.
To the blender with the tomatoes, add reserved chicken broth, chipotle chiles in adobo, onion, garlic, and chicken bouillon. Blend until smooth, about 2 minutes.
To make the tinga:
In a Dutch oven or large saucepan over medium-high heat, heat olive oil until just smoking. And onions and cook until softened, about 5 minutes.
Add half the sauce and bring to boil. Stir in chicken, adding more sauce if desired (the heat of the dish comes from the sauce, so use less sauce for a less spicy dish and more sauce for a spicier dish). Boil until slightly thickened, about 2 minutes.
Remove from heat and serve in tortillas with Mexican rice, shredded lettuce, and cheese.
Chicken: Buy a whole chicken. You make chicken broth with it, which you need for this recipe, then shred up the meat for the tinga. Cut the chicken up into pieces or have your butcher do it for you.
Organ meats: The heart and gizzard can be added to the broth if desired, but the liver should be discarded or reserved for another purpose.
Cold water: Always start with cold water. This helps keep the broth clear, not cloudy.
Chicken broth: It's super simple to make full-blown chicken broth while poaching your chicken. To the pot with the chicken, onion, garlic, and salt, add 1 carrot (peeled and chopped), 1 rib celery (chopped), and a sachet of spices (any or all of these: 6 parsley stems, 1 sprig fresh thyme, 1 bay leaf, 1 teaspoon whole black peppercorns). Cover everything with cold water and proceed with the recipe. After removing and shredding the chicken for the tinga, strain the broth through a fine-mesh strainer or cheesecloth. Reserve 1 cup broth for the sauce. The rest can be covered and chilled in the refrigerator over night. The next day, scrape off the accumulated fat from the top of the broth and discard. Divide the broth into freezer-safe containers (leaving at least 1/2-inch for expansion), label, and freeze.
Chipotle chiles in adobo sauce: If you've never had chipotle chiles in adobo: be warned. It's spicy! If you shy away from spicy food, add just a little to start or omit it entirely. Reconstitute some dried guajillo or pasilla chilies instead, if you prefer, by soak them in warm water for a few minutes, then purée them with a little vinegar. Use leftover chipotle chiles in adobo to make pinto beans or spicy stuffed peppers.
Leftover chicken: To substitute leftover chicken in the fridge, skip the whole part about poaching a chicken and just shred your leftover chicken. Purée all sauce ingredients together in a blender. You won't have the chicken broth from cooking the chicken, so just use store-bought chicken broth or water. You can throw in some bouillon if you want to. Proceed with the rest of the recipe, heating the oil in the pan for the onions, and then adding the sauce.
Help! I made it too spicy: Help is on the way. If it ends up being a little too spicy, serve the tinga with a dollop of sour cream of lime wedges to counterbalance the heat.
Warming tortillas: Heat a dry skillet over medium or medium-high heat (depending on how closely you plan to watch the tortillas). Add a few tortillas in a single layer (try not to overlap) and heat until softened on one side (1-2 minutes). Flip and heat on the other side for another minute or so. Wrap in foil or a tortilla towel, then repeat with remaining tortillas.
Tinga toppings: Cheese, minced onions, cilantro, avocado, hot sauce, shredded lettuce. Add to a bowl with quinoa or Mexican rice or try on a crunchy tostada with beans and guacamole.