Heat olive oil in a large pot or deep skillet over medium heat.
Add the chicken pieces and cook for 5-6 minutes, stirring occasionally, until lightly browned but not fully cooked through. Remove chicken from the pot and set aside.
In the same pot, add the chopped onion and cook for 3-4 minutes until softened.
Add the minced garlic and cook for 1 minute until fragrant.
Add the rinsed rice to the pot and stir to coat with the oil, onion, and garlic mixture.
Pour in the chicken broth, then add salt, black pepper, and dried thyme. Stir to combine.
Return the chicken pieces to the pot, spreading them evenly over the rice.
Bring the mixture to a boil, then reduce heat to low and cover with a tight-fitting lid.
Simmer for 15 minutes without lifting the lid.
After 15 minutes, lift the lid and evenly distribute the broccoli florets over the rice and chicken. Cover again and cook for another 5-7 minutes until the rice is tender, the chicken is cooked through, and the broccoli is bright green and tender-crisp.
Remove the pot from heat and let it sit, covered, for 5 minutes to allow the flavors to meld.
Fluff the rice with a fork and gently mix the broccoli and chicken into the rice before serving.