Heat a large pan to medium heat and add the dried peppers. Dry toast them 1 to 2 minutes per side, until the skins darken. Do not let them burn.
Transfer the toasted peppers to a large bowl, cover with hot water, and steep 20 minutes until softened. Reserve the soaking liquid if you'd like for thinning later as needed.
In the same pan, heat the oil to medium. Add the onion and tomatoes and cook 5 minutes to soften. Add the garlic and cook another minute, stirring.
Add the onion mixture to a food processor or blender. Pull the softened chilies from the soaking liquid and add them, along with the seasonings, apple cider vinegar, and 1 cup chicken stock. Process until smooth. Strain for a smoother sauce.
Season the chicken thighs with salt. Heat the oil in a large Dutch oven to medium-high and sear the thighs 3 to 4 minutes per side, until browned. Work in batches so they aren't crowded. Remove and set aside.
Return the chicken to the pot with all of the birria sauce, the chicken stock, and the bay leaves. Use 2 cups for a thicker consome, 3 cups for a looser one. Bring to a simmer.
Cover and simmer 40 minutes, until the chicken shreds easily with a fork. Keep it at a gentle simmer, not a boil.
Remove the chicken, shred it with two forks, and discard the bay leaves. Return the shredded chicken to the pot and stir to coat. Taste and adjust for salt.
Ladle a cup or so of consome into a shallow bowl. Heat a comal or skillet to medium with a light coat of oil or lard.
Brush a tortilla with consome and lay it in the pan. Add cheese, then chicken lifted out with a slotted spoon. Fold it over and fry 1 to 2 minutes per side, until crisp and the cheese melts. Add a touch more oil between batches as needed.
Top with chopped onion, cilantro, and a squeeze of lime. Serve with hot consome on the side for dipping.