Snip off the chili stems, then cut lengthwise and shake out any seeds. Combine the water and chilies in a medium saucepan and bring to a boil over medium heat. Boil until the chilies are pliable, about 3 minutes.

While chilies are softening, coarsely chop the onion and garlic and transfer to a blender. Use tongs to transfer the boiled chilies to the blender. Add 1/2 cup of the cooking liquid, the tomatoes, and salt, and blend until smooth.

Heat the oil in a large frying pan or Dutch oven over medium heat. Add the guajillo mixture, cover, and cook for 10 minutes, stirring occasionally.

Fold in the tortilla chips a handful at a time, stirring gently until all are evenly covered with sauce. Reduce heat to low, top with the cheese, and cover the pan.

Heat the oil in a second large frying pan over medium heat. Fry eggs to desired level of doneness, about 4 minutes for cooked whites and runny yolks.

Using a wide spatula, arrange the fried eggs on top of the chilaquiles. Remove pan from heat. Slice the radish, finely slice the red onion, and crumble the cotija cheese. Scatter over the chilaquiles. Add the cilantro and season with black pepper. Serve immediately.