Heat a dry skillet over medium heat. Toast guajillo, ancho, and árbol chiles for 20-30 seconds per side until fragrant and puffed. Don't let them burn.
Transfer toasted chiles to a bowl and cover with boiling water. Soak for 20-25 minutes until soft and pliable.
Using the same skillet over medium-high heat, char the tomatoes (cut side down), onion half, and garlic cloves for 4-5 minutes until blackened and softened. Flip and char for another 2-3 minutes.
Drain the soaked chiles, reserving ½ cup of soaking liquid. Add chiles, charred vegetables, vinegar, cumin, oregano, cinnamon, paprika, and 1 cup of chicken broth to a blender. Blend until completely smooth. Strain through a fine-mesh sieve and season with salt and pepper.
Pat the chicken dry and season generously with salt and pepper. Heat oil in a Dutch oven over medium-high heat. Sear chicken skin-side down for 4-5 minutes until golden brown, then flip and sear for 2-3 minutes. Remove chicken and set aside.
Pour the strained chile sauce into the pot, scraping up the browned bits. Add the remaining 3 cups of chicken broth and bay leaves. Nestle the chicken back in, skin side up. Bring to a gentle boil, then reduce heat to low, cover, and braise for 45-55 minutes until tender.
Remove chicken from the consomé and let cool. Discard skin and bones. Shred the meat with two forks and return it to the pot. Remove and discard bay leaves.
Pour consomé into a shallow bowl. Skim 2 tablespoons of fat from the surface and heat it in a large skillet over medium heat.
Dip each tortilla in the consomé, coating both sides. Place in the hot skillet. Add cheese and shredded chicken to one half, then fold over. Press gently and cook until golden and crispy, about 1-2 minutes per side.
Ladle warm consomé into small bowls for dipping. Top tacos with chopped onion and cilantro. Serve with lime wedges. Dip each taco in consomé before eating.