Heat a large skillet and add the olive oil. Add the potatoes and onions to the pan. Sprinkle the herbs de provence on top and cook with out stirring for 5 minutes.
As you are cooking the vegetables add the chicken stock as needed to keep things moving.
Stir the potatoes and onions. Add the green beans and cook for another 3 minutes. stirring occasionally. Add the squashes and cook for 2 minutes or so.
Add the tomatoes, parsley and green onion to the pan and cook for several minutes.
Using a spatula, make four holes in the vegetable mixture. Break one egg into each of the holes. Cover and cook for 2 -4 minutes depending on how hard or soft you like your eggs.
Carefully scoop out the hash into shallow bowls. Top with the grated cheese