In a medium pot over medium heat, poach the whole chicken breasts in just enough room temperature water to cover the breasts, turning heat to low after the water begins to simmer, and adding more water if needed to ensure the chicken is just-covered.
Check the temperature at 8 minutes by poking a meat thermometer into the thickest part of the breast; when the chicken reaches 70°C/158°F remove, set aside, and when cool enough to touch, shred by pulling apart the chicken with your fingers. Keep the poaching liquid for the salsa.
In a deep medium-sized frying pan or skillet over medium heat, add the oil and fry the chopped onion until soft and translucent, add the garlic and continue to fry until fragrant (do not to let the garlic brown), then add the puréed tomatillos, chicken broth and salt and simmer, stirring occasionally, and increasing heat if needed, to reduce and thicken the salsa. Taste and add salt if needed.
Add half the fresh coriander and the shredded chicken to the green salsa, stir to combine well, then add the tortilla chips and continue to simmer until the tortilla chips are just soft; no longer, as you don’t want the tortilla chips to turn to mush.
Distribute the tortilla chips, salsa and shredded chicken between two plates.
Sprinkle on the cotija or crumbly white cheese, remaining fresh coriander, purple shallot slices, and serrano chilli slices or jalapeno slices; spoon on the Mexican crema or sour cream or crème fraiche; and arrange the avocado slices on top, squeezing a little lime juice on them.
Fry the eggs to your liking and pop the eggs on top of the chilaquiles, sprinkle with chilli flakes, and serve immediately with additional dishes of garnishes and hot sauces.