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, sprinkle with 1 teaspoon salt and 1/2 teaspoon pepper, and increase heat to medium-high.
Cook the chicken, stirring occasionally, for 3–4 minutes until the outside is no longer pink but the pieces may not be fully cooked through.
Push the chicken to one side of the pan and move the skillet so the empty side sits over the burner; add 1 tablespoon olive oil to the empty side and let it warm.
Add the uncooked rice to the hot oil and sauté for 2–3 minutes, stirring, until the rice begins to lightly brown.
Pour in the chicken broth, then add the paprika and oregano and stir to combine. Bring the mixture to a boil over medium-high heat.
Once boiling, cover the skillet and reduce heat to medium-low to maintain a gentle simmer; cook covered for 12–13 minutes.
Uncover and add the broccoli, sprinkling the pan with the remaining 1/4 teaspoon salt. Stir to combine with the rice and chicken.
Cover again and cook on low for another 8–9 minutes, until the rice and broccoli are tender and most of the liquid is absorbed.
Turn off the heat, sprinkle the shredded cheddar evenly over the top, cover, and let sit 2 minutes to melt the cheese; then serve.