Preheat the oven to 400 degrees. In a large oven-safe skillet, heat the olive oil on the stove over medium heat.
Add the onion, green beans and broccoli to the pan and cook until the vegetables are tender, about 8 minutes. Add salt and pepper to taste.
Sprinkle the flour over the vegetable mixture and stir continuously until the flour cooks to a blonde color, about 1 to 2 minutes. Slowly stir in the chicken broth and continue stirring until everything is well combined and there are no lumps. Bring this mixture to a simmer and allow it to thicken slightly.
Add the cooked chicken and rice to the pan along with the sour cream and stir to combine. Sprinkle the top with shredded cheese and place the pan in the oven. Cook until the cheese has melted and the casserole is bubbly, about 12-15 minutes.