Poach the chicken breasts in water with a teaspoon of Kosher salt until cooked through. Set aside to cool.
Wipe the pepper off with a damp paper towel and toast the Pasilla pepper – either by holding it with tongs a few inches over an open flame on a gas range or heating it in a dry frying pan at low to medium heat while pressing down on it with a spatula. Heat until the pepper aroma is noticeable, but do not burn the pepper. After toasting, cool, cut off the stem, split the pepper, and remove the seeds and webbing.
Break the toasted pepper into pieces and place in a blender. Add the diced tomatoes, with their juice, to the blender. Puree the peppers and tomatoes and leave in the blender.
Add the oil to a 4-quart kettle and bring to medium heat. Cook the onion and garlic until lightly browned, stirring frequently. Using a slotted spoon, remove the onion and garlic from the oil, straining out as much oil as possible. Place the onion and garlic in the blender with the tomato and pepper mixture, and puree until smooth.
Pour the mixture from the blender into the kettle and raise to medium-high heat. Stir the mixture constantly until it reduces the consistency to nearly that of tomato paste. Add the chicken broth, the Epazote (if used), and 1 teaspoon Kosher salt (to taste). Stir, reduce heat, and simmer for 20 minutes. Before serving, shred the poached chicken breasts, add to the broth, and heat for about 5 minutes – until very hot.
To serve, divide the cheese and avocado into the bowls and ladle the soup into the bowls. Sprinkle on a few tortilla chips and add a dollop of Crema or sour cream to each bowl; sprinkle with fresh, diced cilantro and lime juice, if desired.