Preheat the oven to 375°F (190°C).
Heat olive oil in a large skillet over medium-high heat. Add the chicken pieces and cook until lightly browned but not cooked through, about 4-5 minutes. Remove chicken from skillet and set aside.
In the same skillet, add the chopped onion and cook for 3 minutes until softened.
Add the minced garlic and sliced mushrooms to the skillet and cook for another 4 minutes until mushrooms are tender.
Sprinkle the flour over the vegetables in the skillet and stir well to coat. Cook for 1 minute to remove the raw flour taste.
Gradually whisk in the chicken broth and milk, stirring constantly until the sauce thickens, about 3-4 minutes.
Add salt, black pepper, and dried thyme to the sauce and stir to combine.
Return the chicken pieces to the skillet along with the green beans. Stir to coat everything evenly with the sauce.
Transfer the mixture to a 9×13-inch baking dish. Sprinkle the shredded cheddar cheese evenly over the top.
In a small bowl, mix the breadcrumbs with the melted butter until combined. Sprinkle the breadcrumb mixture evenly over the cheese layer.
Bake in the preheated oven for 25 minutes, or until the casserole is bubbly and the breadcrumb topping is golden brown.
Remove from the oven and let it rest for 5 minutes before serving.