Preheat the oven to 375 degrees F (190 degrees C).
Heat 1 1/2 tablespoons oil in a large skillet over medium heat. Fry tortillas, one at a time, until golden and lightly crisp, about 30 seconds per side. Drain on paper towels and set aside.
Add ham to the skillet and cook until lightly browned, about 1 minute. Transfer ham with a slotted spoon to a paper towel-lined plate to drain; set aside.
Heat remaining 1 1/2 tablespoons oil in the skillet. Add onion and 1/2 teaspoon salt; cook, stirring often, until onion is translucent. Add beans and green chiles; cook, stirring and partially mashing, for 1 minute. Remove from heat.
Spray a 9×13-inch baking dish liberally with cooking spray. Place tortillas in a single layer on the bottom. Sprinkle ham and cheese over the tortillas. Top evenly with the black bean mixture and pico de gallo. Form 6 shallow wells in the pico layer; crack an egg into each well.
Bake in the preheated oven until egg whites are just set and yolks are runny, about 20 minutes. Sprinkle with pepper and remaining 1/4 teaspoon salt. Top with avocado and cilantro.
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