In Mexico, this tortilla chip dish is known as “divorced” chilaquiles because of the contrasting red and green salsa toppings. To make it more substantial, chef Enrique Olvera adds shredded chicken and fried eggs. Slideshow: More Egg Recipes
In a medium saucepan, combine all of the ingredients, season with salt and bring to a simmer. Cook over moderately low heat, stirring occasionally, until saucy, about 25 minutes; keep warm.
In another medium saucepan, cover the chicken, garlic and bay leaf with 1 inch of water. Bring to a simmer and cook over low heat until the chicken is white throughout, about 20 minutes. Transfer the chicken to a plate to cool slightly; reserve the cooking liquid for another use. Coarsely shred the chicken and season with salt; keep warm.
In a large nonstick skillet, heat the olive oil. Crack the eggs into the skillet and season with salt. Cover and cook over low heat until the whites are firm but the yolks are still a little runny, about 3 minutes.
Mound the tortilla chips on plates. Spoon some of the salsa roja over half of the chips and some salsa verde over the rest. Top with the shredded chicken and fried eggs. Garnish with cheese, sour cream and onion and serve with the remaining salsa roja and salsa verde.
The salsa roja and shredded chicken can be refrigerated separately for 3 days. Rewarm both before serving.