Add olive oil over medium heat to a large skillet. Season chicken with salt and pepper and add to the skillet. Sauté until the chicken is cooked through, 7-10 minutes. Remove the chicken to a plate and set aside.
Add mushrooms, celery, and onion to the skillet. Cook, stirring occasionally, until the vegetables are softened, about 7 minutes. Add garlic and thyme; stir for one minute.
Sprinkle with flour and stir to combine. Stir in half-and-half; bring to a heavy simmer. Simmer, stirring frequently, until the sauce thickens. Add salt and pepper to taste
Stir green beans and chicken into the sauce. Transfer to a 2-quart casserole dish.
Toss grated parmesan cheese with panko bread crumbs and sprinkle over the casserole. Bake until golden and bubbly - 15 to 20 minutes. Let stand 5 minutes before serving.