Stem and seed the chiles, reserving 1 tablespoon of the seeds, or more to taste.
Toast the chiles in a large cast iron skillet over high heat. Transfer the chiles to a bowl, cover with hot tap water, and soak for 20 minutes. Drain.
Meanwhile, toast the chile seeds in the skillet. In a clean coffee or spice grinder, finely grind the chile seeds, sesame seeds, peppercorns, cloves, and cinnamon.
In a blender, combine the chiles, ground spices, 1/2 cup chicken broth, and garlic and blend.
Heat the lard in a saucepan over medium-high heat. Pour in the blended sauce and cook, stirring and scraping the bottom, until somewhat reduced, about 10 minutes. Add the remaining 3 cups chicken broth, bring to a boil, lower the heat and simmer until thickened, about 20 minutes more. Season with salt and pepper. (The sauce may be prepared up to 3 days ahead and stored in the refrigerator.)
Preheat a grill or grill pan. Lightly brush the chicken with oil and season with salt and pepper. Grill the chicken, turning to make cross-hatched marks, until just cooked through, about 12 minutes. Transfer the chicken to a platter and let rest for 5 minutes.
Slice the chicken and transfer to a platter.
Reheat the sauce and stir in the avocado leaf. Ladle the sauce over the chicken, sprinkle with sesame seeds, and garnish with herb leaves. Serve with corn tortillas.
Serving Suggestion: Warm corn tortillas.