In a large saucepan, heat the oil over medium heat. Add onion and cook until it starts to soften. Nudge onions to one side to make room for the peppers, and continue cooking until the onion takes on a golden sheen. Add the garlic to the onion, and make room for the mushrooms. Cook until mushrooms are soft.
Using a slotted spoon, lift all the vegetables from the pan into a bowl, keeping any remaining oil in the pan.
Add ham to the pan, and sauté for several minutes until the slivers sear just a bit. Add ham to vegetables in bowl.
Into the now almost-dry pan, stir-fry broccoli for a minute. Add white wine or water, then cover to steam for another minute. Add to ham and vegetables. Cover to keep warm.
Pour pasta into the saucepan and cover with 4 cups water. Bring to a boil, then reduce to a steady simmer, stirring occasionally. When pasta is done, carefully drain any remaining water into a cup. Add the reserved vegetables and meat, along with the cheese, and heat through, adding pasta water as needed to prevent sticking. Season to taste and serve at once.
Nutrition information per serving: