Preheat oven to 400°F. Scrub potato, then pierce several times with a fork. Rub the skin with olive oil and a bit of salt. Place directly on the middle rack of the oven, with a pan underneath to catch any drippings. Bake for 1 to 1-1/2 hours, until cooked through and soft.
Let the potato cool slightly, then cut potato in half lengthwise. Use a spoon to scoop out the insides of each half, leaving about 1/4” layer of potato on the skin. Transfer the scooped out potato to a small bowl and mash with a fork.
Scramble eggs in a non-stick skillet until almost done but still slightly wet. Transfer the eggs to the bowl with the mashed potato. Add cheese and season to taste with salt and pepper, then mix thoroughly.
Spoon the mixture into the shelled out potato skins. Heat potatoes in a microwave on high for 30 seconds to 1 minute, until potatoes are hot, cheese is melted, and eggs are completely cooked. Spoon salsa on top, then serve (with additional salsa on the side, if desired).