Place the flour in a swallow dish. Pat the chicken dry then season on both sides with salt and pepper. Dredge the chicken in the flour and shake off the excess.
Heat 2 tablespoons oil in a 12-inch skillet over medium-high heat until shimmering. Brown the chicken on one side, about 4-5 minutes. Transfer to a plate and set aside. You aren't trying to cook the chicken through on this step, but rather just giving it a little color/flavor.
Add the broccoli florets to the now empty pan (adding a bit more oil if needed) along with a pinch of salt and cook briefly, for about 3-4 minutes. Transfer the broccoli to a bowl.
Add the remaining tablespoon of oil the pan along with the minced onion and ½ teaspoon salt. Cook until softened, about 5 minutes. Stir in the rice and garlic and cook until fragrant, about 30 seconds.
Increase the heat to high and pour in the chicken stock. Nestle the chicken into the rice, browned side up. Bring to a boil, then cover and reduce to a simmer over low heat. Cook covered for 12-15 minutes.
When the rice is done cooking, stir in the broccoli, ½ cup of the cheese and the Tabasco sauce. Combine evenly then smooth the top and sprinkle with the remaining cheese. Turn off the heat, cover the pan and let sit until the cheese melts and the broccoli is warmed, about 5-10 minutes. Enjoy!