Heat a 12-inch cast-iron skillet over medium heat. Add the chorizo, breaking it up with a wooden spoon, and cook for 6 to 8 minutes until browned and cooked through with no pink remaining.
Push the chorizo to one side. If the pan looks dry, add the olive oil, then add the onion and bell pepper. Cook for 4 to 5 minutes until softened.
Stir in the garlic, cumin, and chili powder. Cook for 30 seconds until fragrant, stirring so it doesn't scorch.
Add the black beans and stir everything together. Cook for 2 to 3 minutes until the beans are heated through. Season with salt and pepper.
Use the back of a spoon to make 8 shallow wells in the mixture. Crack one egg into each well.
Cover the skillet, reduce heat to low, and cook for 6 to 8 minutes until the egg whites are fully set. Check at 6 minutes for a soft yolk, closer to 9 minutes for a fully set yolk.
Sprinkle the shredded cheddar or Mexican blend cheese over the top, avoiding the yolks. Cover for 1 to 2 more minutes until melted.
Remove from heat and scatter cilantro over the top. Serve straight from the skillet with lime wedges.