Best Breakfast Potatoes with Apple

olive oil

to 3 medium potatoes (washed but unpeeled)

medium onion (yellow or white)

carrots (washed but unpeeled, optional)

mushrooms (about half of an 8 ounce package)

red or yellow bell pepper (optional)

red chili flakes (more or less to taste)

paprika

to 1 teaspoon kosher or sea salt (to taste)

to 1/2 teaspoon ground pepper (to taste)

medium apple (a tart apple works best, but any will do)

eggs (organic, pastured)

shredded cheese (cheddar, jack, or mozzarella all work well)