Stir the tomatoes, white onion, cilantro, lime juice, and salt together in a medium bowl, then set the pico de gallo aside. The mixture should look juicy but fresh, with the salt starting to draw out a little tomato liquid.
Heat the olive oil in a small saucepan over medium heat, then add the onion and salt and cook for 3 to 6 minutes, stirring now and then. Stop when the onion looks soft and translucent and the pan smells sweet rather than sharp.
Add the cumin and cook for about 30 seconds, then stir in the beans and water, cover, and cook for 5 minutes. You should smell the cumin bloom quickly before the beans warm through and soften.
Lower the heat, uncover the pan, and mash at least half the beans with a potato masher or the back of a fork, then cook 2 to 3 minutes more, stirring often. The beans should become thick and creamy while still keeping some texture.
Take the beans off the heat and stir in black pepper and lime juice, adjusting with more salt, pepper, lime juice, or a tiny splash of water if needed. They should spread easily without running and stay warm under a cover until serving.
Warm the salsa in a medium saucepan over medium heat until it reaches a gentle simmer, then keep it on low. The sauce should be hot and lightly bubbling, not boiled down too far.
Warm each tortilla one at a time in a small skillet over medium heat, turning as needed, then spread each with beans and place on its own plate, adding Monterey Jack if using. The tortillas should be soft and flexible, not stiff or crisp.
Using the same skillet over medium heat, add 1 teaspoon olive oil and cook one egg at a time, repeating with the remaining oil and eggs. The whites should be set and the yolk cooked to your preferred doneness, with lightly crisp edges if the oil is hot enough.
Set an egg on each prepared tortilla, spoon warm salsa over the dish without covering the yolk, then add pico de gallo with a slotted spoon or fork and finish with black pepper and any optional garnishes. Keep the extra tomato juices behind so the top stays bright and fresh instead of watery.
Serve right away while the eggs, beans, and salsa are still hot. The tortilla stays at its best texture when the plate goes straight from assembly to the table.