Position racks in the upper and lower thirds of the oven and preheat to 425 degrees F. Bring the chicken broth to a low simmer in a saucepan. Toss the broccoli with the olive oil on a rimmed baking sheet.
Melt 2 tablespoons butter in a large Dutch oven or ovenproof pot over medium-high heat. Add the onion and cook, stirring occasionally, until slightly softened, about 2 minutes. Add the rice and stir to coat. Pour in the wine and cook until evaporated, about 1 minute. Add the hot broth, 3/4 teaspoon salt, and pepper to taste; bring to a boil. Cover and set on the bottom oven rack. Place the broccoli on the upper rack. Bake, stirring the rice and broccoli once halfway through cooking, until most of the liquid has been absorbed in the rice and the broccoli is tender, 20 to 25 minutes.
Remove the rice and broccoli from the oven. Add 3/4 cup hot water, the remaining 1 tablespoon butter and the cheese to the rice and stir until creamy (add a little more hot water to loosen, if necessary). Stir in the broccoli.