At the same time, oil a large, heavy bottom pan and set it over medium-high heat. Add chorizo, and break it up while stirring continuously, until the meat is cooked through. Drain, and set aside. (see notes for best draining method)
To that same pan, add olive oil, and set over medium heat. Add onion, green pepper, jalapeño(s), and garlic. Sauté until the onions become translucent, about 5 minutes.
Add black beans and a pinch of salt. Stir to combine, and allow the mixture to cook for 2-3 more minutes.
Stir in the salsa and drained chorizo, and bring the mixture to a simmer. By this time, your potatoes should be browned and fork-tender. Add them to the chorizo mixture, and stir to evenly distribute.
Dig 4 wells into the mixture, so the eggs have a place to sit. Crack an egg into each well, and cover. Allow the eggs to poach in the residual liquid for about 3 minutes, similarly to poaching an egg in water. When the egg is done, the yolk will be runny and the whites will be firm.
Uncover and remove from heat. Serve with avocado and cheese garnish! For extra flair, serve on top of toasted corn tortillas.