Heat the olive oil in your largest nonstick pan with a lid on high heat. Add the potatoes and toss to coat with the oil. Cover, cook 2 minutes, then lift the lid and stir to keep the potatoes from sticking to the pan. Repeat until the potatoes just start to soften (~10 minutes). If necessary, reduce the heat to prevent burning.
Add the garlic, red bell pepper, carrot, ham, and a couple good pinches salt and pepper. Repeat the cover-cook-toss every 2 minutes for 6-8 minutes until the potatoes are cooked through.
Remove from the heat and toss in the tomato, parsley, and green onion, reserving some onion for final garnish if desired.
Heat a nonstick pan spray with cooking spray over medium heat, and cook the eggs over easy. Season with salt and pepper.
Serve the potato hash with an egg on top, garnished with fresh cracked pepper, green onion, and parsley. Yields 6 servings.