Place chicken on a plate and sprinkle with rosemary and flour and toss to coat.
Heat 1 teaspoon oil in a large nonstick skillet over medium-high heat. Add chicken; cook, turning regularly with tongs, until lightly browned, and chicken pieces are cooked through.
In same skillet over medium-high heat, heat remaining 2 teaspoons oil. Add onion and garlic; sauté until lightly browned and almost tender, about 4 minutes.
Add the chicken broth and bring to a simmer, stirring bottom of pan to scrape up browned bits, about 1 minute.
Add the broccoli and chicken back into the pot. Cover with a lid and cook for 5-7 minutes, or until broccoli is tender. Sprinkle with salt and pepper to taste.
Serve over cooked brown rice, if desired.