For the chorizo: Melt the lard in a medium saucepan over medium heat, then add the achiote, chili powder, cumin seeds, jalapeno and oregano and toast until fragrant, about 2 minutes. Cool slightly, then add to the pork in a bowl and mix well.
Heat a cast-iron skillet over medium-high heat and brown the seasoned pork, stirring often, 5 to 7 minutes. Remove the pork, reserving the pork fat in the skillet.
Add the red onion to the skillet and adjust the heat to medium. Cook until soft and lightly caramelized, 6 to 8 minutes. Add the pork back to the pan, followed by the eggs. Scramble the eggs into the "chorizo," then add the cheese and mix thoroughly until melted. Set aside.
For the tomatillo sauce: Preheat the oven to 450 degrees F.
Toss the tomatillos, garlic, onion, lime juice, half of the oil and a pinch of salt in a mixing bowl, then transfer to a rimmed baking sheet and roast until tender and charred, about 20 minutes. Add to a blender along with the vinegar, remaining 2 tablespoons oil and a pinch of salt. Blend until smooth, then set aside.
For the potatoes: Dice the potatoes into 1-inch cubes. Bring a medium pot of salted water to a boil, then add the potatoes and cook until almost fork-tender, about 7 minutes. Drain and set aside to cool slightly.
Meanwhile, clip a deep-frying thermometer to a large, heavy-bottomed skillet. Heat 1 1/2 inches vegetable oil to 375 degrees F.
Fry the potatoes until golden, 2 to 3 minutes. Remove with a slotted spoon and drain on paper towels.
For the refried beans: Meanwhile, puree the black beans and lard in a food processor until smooth. Heat in a saucepan until warm.
Heat the tortillas in a hot dry skillet or directly over a flame until pliable. Place a spoonful of the beans on each tortilla, then cover with the potatoes, then egg-chorizo mixture. Drizzle some the tomatillo sauce on top. Roll and serve.