Heat a cast iron skillet over medium heat and add olive oil. Add the sausage to the skillet and break it into small crumbles as it cooks.
When the sausage is almost fully cooked, add the chopped peppers and onions to the skillet. Continue cooking until the sausage is fully browned and the vegetables are tender.
Remove the sausage, peppers, and onions from the skillet and set them aside in a bowl.
Add more olive oil to the skillet if needed. Once heated, add the diced sweet potatoes to the pan. Sprinkle with salt, pepper, chipotle powder, garlic powder, and paprika. Cook the potatoes, stirring occasionally, until they are browned on all sides and tender. This should take about 10–15 minutes.
Return the cooked sausage, peppers, and onions to the skillet with the sweet potatoes. Stir well to combine and heat through.
In a separate skillet, prepare the eggs to your liking. For this dish, over-easy or sunny-side-up eggs work best, allowing the runny yolk to act as a sauce for the hash.
Plate the sweet potato hash and top each serving with a freshly cooked egg. Enjoy!