Heat a large saucepan over medium heat for 4-5 minutes.
When the pan is hot, add the oil.
When the oil shimmers, add the onion and garlic along with a heavy pinch of salt and freshly ground white pepper.
Cook, stirring occasionally, until the onions begin to turn golden.
Stir in the wine and scrape the bottom of the pan, allowing the wine to cook down until dry.
Add the stock or broth and bring to a boil.
Turn down the heat to medium, and add the pasta. Cook for 3-4 minutes.
Add the broccoli and continue to simmer until the broccoli and pasta are tender, another 5-7 minutes.
Taste and adjust the seasonings. If desired, top each bowl of soup with some freshly grated cheese.