Heat 2 teaspoons of the olive oil in a large skillet over medium‑low heat. Add the diced onion and minced garlic and sauté until softened, about 1–2 minutes.
Add the bite‑sized chicken pieces to the skillet and sprinkle 1 teaspoon of salt and 1/2 teaspoon of pepper over them. Increase heat to medium‑high and cook the chicken pieces, stirring occasionally, until the exterior is no longer pink, about 3–4 minutes.
Push the chicken to one side of the skillet and return the pan so the empty side is over the burner. Add 1 tablespoon of olive oil to the empty side and heat briefly.
Add the uncooked long‑grain rice to the oiled side and sauté, stirring, for 2–3 minutes until it begins to brown slightly.
Stir the chicken and rice together, then add the chicken broth, paprika, and oregano. Increase heat to medium‑high and bring the mixture to a boil.
Once boiling, cover the skillet and reduce heat to medium‑low to maintain a gentle simmer. Cook covered for 12–13 minutes.
Remove the lid, add the broccoli florets and sprinkle the remaining 1/4 teaspoon salt over the pan. Stir to combine the broccoli with the rice and chicken.
Cover again and cook on low for another 8–9 minutes, or until the rice and broccoli are tender and most of the liquid is absorbed (there may be a small amount of liquid remaining).
Turn off the heat, sprinkle the shredded cheddar evenly over the top, cover for 2 minutes to melt the cheese, then serve.