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.