Don't rush the onions: high heat browns the outside before they turn sweet. Chop them or slice them thinly, as you prefer. The brown sugar is optional, and a teaspoon is enough. Traditional Parmigiano Reggiano is made with animal rennet, so choose a vegetarian hard cheese if you eat strictly vegetarian. Vegan: use vegan butter or extra olive oil and 2 to 3 tablespoons of nutritional yeast instead of parmesan. The caramelized onions keep in the fridge for up to 4 days.