Place the cut potatoes in a medium pot, cover with water, and bring to a boil over high heat. Boil for 5 minutes, drain, and set aside.
In a large, heavy, oven-proof skillet (a 12" cast iron is ideal) heat the olive-oil over medium high heat. Add the potatoes and cook until golden brown, turning once. Add the ham, green onions and asparagus, and sauté over medium heat for another 2 minutes.
Whisk together the eggs, salt, cheese, and pepper. Pour over the potatoes directly into the pan, shake the pan a couple times gently to distribute the egg, and place the pan in the oven for 20-25 minutes, until the egg is set and the top is golden brown.
Remove from the oven, and run a knife along the inside perimeter to loosen from the pan. Allow to cool in the pan for 10-15 minutes, before inverting onto a serving plate. Cut in slices and serve.
This recipe uses Frescolio's Basil Infused EVOO. You could also try it with any of the following EVOOs for a different taste: