Heat the oil in a large saucepan over medium-high heat until shimmering. Add the onions and brown sugar and cook, stirring occasionally, until the onions are deeply caramelized and dark brown, about 25 to 30 minutes. If the onions begin to stick, add a tablespoon of water at a time and scrape the pan to release the flavor.

Stir in the garlic and red pepper flakes and cook until fragrant. Pour in the wine and simmer until the liquid has mostly evaporated, scraping the bottom of the pan to incorporate all the browned bits. Spread the onion mixture into an even layer in a 2-quart baking dish.

Adjust the oven rack to the middle position and heat the oven to 400 degrees. Use the same saucepan to combine the beans, broth, and rosemary. Simmer for 10 minutes, which jump-starts the cooking and allows the herbs to infuse the beans.

Remove the pan from the heat and stir in the Parmesan. Season with salt and pepper to taste. Pour the bean mixture over the onions in the baking dish and sprinkle the Gruyère evenly over the top.

Bake until the cheese is golden brown and the sauce is bubbling at the edges, about 20 minutes. Let the gratin rest for 10 minutes so the sauce can thicken, then sprinkle with parsley and serve warm.