Preheat the oven to 350ºF. Generously spray a muffin tin with olive oil or nonstick cooking spray. Set aside.
Add the black beans to a medium bowl. Use a fork or potato masher to roughly mash the beans. Add in the jalapeño, green onion, lime juice, chili powder, cumin, salt and cayenne pepper. Mix to combine (it will be a thick mixture). Taste and adjust any seasonings as needed.
With the pointed ends facing down, press 3 tortilla pieces into each muffin tin, arranging them so that the sides overlap and they form a cup (leaving a lip at the top so the cup is slightly taller than the tin). Spoon 1 ½ to 2 teaspoons of the bean mixture into each cup (smearing it down), followed by some salsa and cheddar cheese. Carefully crack one egg into each of the assembled cups, then transfer to the oven. Bake for about 25 to 30 minutes, until the eggs are set and cooked to your liking.
Let the cups rest for about 5 to 10 minutes, then remove from the muffin tin (use a butter knife to loosen the edges if needed). Serve as desired.
There will be leftover bean mixture after making the breakfast cups; it can be used as a dip, a topping for nachos, filling for burritos or quesadillas, etc.