In a small pot, combine beans with water to cover by 1 1/2 inches. Bring to a boil, reduce to a simmer, and cook until tender, about 45 minutes. Do not drain.
Meanwhile, in a skillet, heat 1 tablespoon oil over medium-high heat. Add chopped onion and cook until translucent, about 7 minutes. Add garlic and cumin and cook 1 minute. Add tomatoes and chiles. Simmer salsa, partially covered, 10 to 15 minutes. Stir in lime juice and salt to taste; keep sauce warm. Preheat the oven to 250°F.
In a skillet, melt lard over medium-high heat. Add minced onion and cook until translucent. Add beans and bean broth gradually, mashing to a chunky paste with each addition. Cook until liquid is mostly evaporated, 8 minutes. Add salt and keep warm.
In another skillet, cook chorizo over medium-high heat, until browned and cooked through. Drain on paper towels and keep warm.
Wipe skillet and add 1/2-inch oil. Quickly fry tortillas one at a time until crisp and lightly browned, placing in oven to keep warm.
Pour oil out of skillet, lightly cover bottom. Over low heat, crack two eggs into skillet and cook to desired doneness; season with salt and pepper and place in oven to keep warm. Repeat with remaining 6 eggs.
To serve arrange one tortilla and top with two eggs. Spoon salsa around yolks and arrange beans, chorizo, rice and avocado alongside. Top with cheese.