Heat the butter in a large skillet over medium high heat. Swirl to coat the pan.
Add the potatoes and let them cook, stirring occasionally until starting to brown. Add the onion, pepper, and a bit of salt and let cook until soft, about 5 yo 7 minutes longer. If the potatoes haven't gotten nice and golden yet, just let them work a little longer.
When the potatoes are looking perfect, stir in the tomato paste and cumin and let cook for a minute until everything is rust colored.
Stir in the salsa and let thicken a minute or two. Finally, stir in the beans.
Make 4 little holes in the hash and gently crack and egg placing one in each spot. Allow to cook until whites are set. Immediately remove from heat and sprinkle with cheese.
Serve with a tosted slice of hearty multi-grain bread, additional salsa, light sour cream and sliced avocado for a full on breakfast fiesta.