In a medium saucepan, melt the butter over medium heat. Add onions and cook, stirring frequently, for 3 minutes. Add the broccoli and cook for another 3 minutes. Add the rice and cook, stirring constantly, for 1 minute. Stir in the broth. Increase the heat to medium-high and bring the mixture to a boil. Cover, reduce the heat and simmer until the rice is tender, 15 to 20 minutes.