This is one of those soups that relies on the caramelization of the onions. Don't rush it, accept the fact that it takes time and depending on the sugar content of the onions may take even longer. Obviously this recipe is adapted from Marcella Hazan's in her book Marcella Cucina. —thirschfeld
unsalted butter, divided
extra virgin olive oil
yellow onion, julienned
Kosher salt and freshly ground black pepper
garlic cloves, peeled and minced (about 1 tablespoon)
In a 3 1/2-quart heavy-bottomed pot, combine the olive oil and half the butter. Place the pot over medium heat. Once the butter begins to melt, add the onions. Season them with a pinch of salt and freshly ground black pepper.
Saute the onions until they become golden. Don't rush this step and adjust the heat as necessary to keep them from browning too fast. Add the garlic and cook until fragrant.
Add the potatoes. Stir them to coat with oil and let them sizzle away for a minute or two. Add the broccoli and do the same as you did with the potatoes. Add the stock.
Bring the stock to a boil. Taste the broth and adjust the seasoning. Go easy on the salt though because the Parmesan has lots and will act as seasoning as well.
Simmer the soup until the broccoli and potatoes are tender. The broccoli is not going to remain vibrant green, but if it is good broccoli it won't be olive drab either.
Once the potatoes have cooked through, add the parmesan, the remaining butter, and the basil. Stir to combine and serve with more black pepper.