Adjust an oven rack to the middle position and heat the oven to 375°F. Heat the vegetable oil in a large Dutch oven over medium-high heat until the oil shimmers but does not smoke.

Add the onions and sugar. Cook, stirring frequently, until the onions have softened and released their liquid, about 5 minutes.

Reduce the heat to medium-low and continue to cook, stirring often, until the onions are dark golden and deeply caramelized, about 20 to 25 minutes.

Stir in the garlic and pepper flakes and cook until fragrant. Pour in the wine and simmer, scraping up any browned bits from the bottom of the pot, until the liquid has nearly evaporated, about 1 minute.

Transfer the onion mixture to a 13 by 9-inch baking dish and spread it into an even layer.

Add the beans, broth, and rosemary to the now-empty Dutch oven and bring to a brief simmer over medium heat, about 1 minute.

Remove the pot from the heat. Gently stir in the extra-virgin olive oil and Parmesan, taking care not to mash the beans. Season with salt and pepper to taste.

Spread the bean mixture evenly over the onions in the baking dish, then sprinkle the Gruyère over the top.

Bake, uncovered, until the cheese is lightly golden and the sauce is bubbling around the edges, 15 to 20 minutes.

Sprinkle with parsley and serve warm, while the cheese is still melted and gooey.