Preparation

• Heat the olive oil in a 5-quart Dutch oven or other heavy-duty pot over medium-high heat until shimmering hot. Add the onions and reduce the heat to medium. Cook without stirring until the bottom of the pot begins to brown, 5 to 7 minutes. Sprinkle with 1/4 tsp. salt and stir with a wooden spatula. Continue to cook, stirring and scraping the bottom of the pot frequently and adjusting the heat as necessary, until the onions are well browned, 20 to 30 minutes.
• Meanwhile, in a 3-quart saucepan, combine the broth with 2 cups water and set over medium heat until steaming hot.
• Transfer the onions to a small bowl, cover, and keep warm. Add about 1/2 cup of the broth mixture to the pot. Scrape the bottom of the pot with the wooden spatula until any stuck-on bits are released. Pour the liquid back into the broth mixture. Wipe out the pot.
• Melt 2 Tbs. of the butter in the pot over medium heat. Add the rice and cook, stirring gently, until glossy and translucent around the edges, about 1 minute. Add the wine and cook, stirring, until absorbed, about 1 minute. Ladle enough broth over the rice to just cover (about 1/2 cup) and simmer, stirring often, until most of the broth has been absorbed (test by running a spoon through the rice; no broth should pool on the pot bottom). Continue to add the broth in this manner, stirring often, until the rice is tender but still has a bit of resistance when you bite into it, 20 to 30 minutes. (You may not need all of the broth.)
• Stir in the remaining 2 Tbs. butter, the cheese, and all but 1/3 cup of the onions. Season with salt and pepper to taste and serve immediately, topped with the remaining onions.