Preheat the oven to 400°F (200°C).
Toss the chipotle chile peppers in a smallish bowl and add just enough boiling water to cover. Let the chipotles soak for about 30 minutes.
Meanwhile, place the tomatoes, cut-side up, in a single layer in a large roasting pan or rimmed baking sheet lined with aluminum foil. Drizzle with oil, season with salt and pepper, and sprinkle with sugar. Roast until the tomatoes begin to collapse and shrink and are slightly charred in places, about 1 hour. Let cool.
While the tomatoes are roasting, peel the casing from the chorizo and discard. Chop or gently break up the meat. In a large skillet set over medium heat, warm 1 tablespoon oil. Add the chorizo and cook, stirring occasionally, until the chorizo is well colored all over, 4 to 5 minutes. Use a slotted spoon to transfer it to a Dutch oven or pot, leaving the fat in the skillet.
Increase the heat under the skillet to medium-high and add some of the pork to the skillet, taking care not to crowd the meat in the skillet, about 5 minutes. Transfer each batch of pork to the Dutch oven as it’s ready, leaving any fat in the skillet.
Toss the onions in the same skillet with the drippings over medium heat, adding a couple teaspoons of oil if the skillet seems dry. Cook the onions, stirring occasionally, until they’re dark golden, about 10 minutes. Add the garlic and cumin and cook for 2 minutes more, then add this to the pork along with the thyme, oregano, salt, and pepper.
Pour a little of the stock or water into the skillet and cook, stirring constantly, for 1 to 2 minutes to deglaze it. Pour this into the Dutch oven along with the rest of the stock or water.
Separate the chipotles from their soaking liquid, reserving the liquid. Remove and discard the stalks and some of the seeds (or you can leave some in if you want a little more heat), and chop the chipotles. Add the chopped chipotles and their soaking liquid to the pork.
Bring the pork to a gentle boil and immediately reduce the heat to low. Partially cover the pot and gently cook until the pork is tender and the tinga is thick, about 90 minutes. Halfway through the cooking time, coarsely chop the tomatoes and add them and any juices on the baking sheet to the pork. (Alternatively, if you don’t care to babysit the tinga on the stovetop, you can transfer the covered pot to a 350°F [176°C] oven until the pork is tender and the tinga is thick, about 90 minutes.)