Preheat the oven to 350 degrees. In a large bowl, season the chicken with salt and pepper, and toss with flour to coat thoroughly. Let the chicken stand for a few minutes, until the coating gets soggy. Pour 3 tablespoons of the vegetable oil over the chicken and toss to coat, then coat the chicken with the sesame seeds.

In a large nonstick skillet, heat the remaining 1 tablespoon of oil. Add the chicken in an even layer and cook over high heat, undisturbed, until browned on the bottom, about 3 minutes. Turn the chicken and cook until browned on the second side, about 2 minutes. With tongs, transfer the chicken to a rimmed baking sheet and put it in the oven while you finish the dish.

In the oil remaining in the skillet, cook the ginger, garlic, and crushed red pepper over moderately high heat until fragrant, about 2 minutes. Add the chicken broth and boil over high heat until reduced by half, about 4 minutes. Add the broccoli, cover, and cook until bright green and crisp-tender, about 2 minutes. Stir in the oyster sauce, season with salt, and remove from the heat. Gently stir in the chicken and sesame oil and serve with steamed rice.