Heat oven to 350 F.
Heat a large skillet over low heat and add 2 teaspoons of olive oil. Add sliced onion and saute for 15 to 20 minutes, stirring frequently, until onions are golden. Remove onions from skillet and set aside.
Add the remaining 1 teaspoon of olive oil, chopped onion and garlic to skillet and cook for 2 to 3 minutes. Add water and mushrooms, cooking for an additional 5 minutes. Sprinkle flour and thyme over mixture and stir. Gradually stir in the milk. Increase heat to medium and stir constantly until the sauce thickens.
To cook the green beans, boil for 8 to 10 minutes in water. (Or, to save time, cook the beans with 2 tablespoons of water, covered in the microwave on high for 5 minutes.)
Drain beans and set aside. Spray a 2-quart casserole with cooking spray and place the green beans in the casserole. Pour the mushroom sauce over the green beans. Top with the sauteed onion slices and fresh bread crumbs. Bake for 15 minutes, or until the top is golden brown.