Heat the olive oil in a large skillet over medium heat. Once heated, add the potatoes, season with salt and pepper, and let cook until almost fork tender, about 5 minutes. Add the garlic, Brussels sprouts and white beans, stir together, and season with smoked paprika, chili powder, salt, and pepper. Cook, stirring often, until sprouts are browned and potatoes are completely fork tender, about 10 more minutes, adding vegetable broth by the tablespoon as the vegetables dry out and potentially stick to the bottom of the pan.

Meanwhile, fry four eggs in a large skillet or in batches. Divide the hash onto plates and top with the fried eggs. Serve with extra cracked pepper.