Heat 2 tablespoons olive oil and 2 tablespoons butter in a large heavy bottom pot or dutch oven over medium heat.
Add in cubed chicken and season with 1/2 teaspoon salt and 1/2 teaspoon black pepper. Cook chicken for 5 minutes until exterior gets some browning.
Add in 1 cup diced white onion and cook alongside the chicken until the onions are translucent, about 3-5 minutes, stirring frequently.
Add 1 1/2 cups long grain white rice and 1 tablespoon minced garlic and cook for 2-3 minutes, stirring constantly, until the rice is lightly toasted. Add in 3 tablespoons all-purpose flour and stir until all of the flour has been absorbed., about 1 minute.
Slowly pour in 2 1/2 cups chicken broth while stirring continuously until the sauce thickens, about 2 minutes. Stir in 2 cups milk, 4 cups broccoli florets, 1 teaspoon dried thyme, and 1 teaspoon dried oregano.
Bring this to a boil and then turn the heat down to medium-low. Cover with a lid and cook until the rice is almost cooked, about 15-20 minutes.
Stir in 1 cup cheese. Top with the remaining cheese and place the lid back on for 2 to 3 minutes until the cheese has melted.