Make pickled moritas. In a small saucepan, combine the piloncillo (or brown sugar), vinegar and 1/3 cup water. Bring to a simmer and stir until the piloncillo dissolves. Add the garlic, chiles and ½ teaspoon salt, then simmer for a couple of minutes. Cool. You’ll want to weight the mixture with a small plate to keep the chiles submerged. Let stand for a couple of hours and let the sweet-sour liquid to soak into the chiles until they're softened.
Make the chicken tinga. Set a very large (12-inch) skillet over medium-high heat and film the bottom liberally with oil. When hot add the onion. Stir for several minutes until it softens, then lay the chicken over the pan. Nestle it down to the bottom of the pan so that it will brown. Cook until everything is richly browned, then stir in the garlic and cook a minute longer.
In a blender or food processor, process the tomatoes to a very coarse puree. Add to the browned chicken in the skillet, along with the chiles and oregano. Reduce the heat to medium-low and simmer until the chicken is tender and the mixture is thick. Taste and season with salt.
Serve. Serve your chicken tinga on warm tortillas with slices of avocado, a sprinkling of fresh cheese and a few cilantro leaves.