Preheat the oven to 350°F and lightly grease a 13x9-inch baking dish.
Bring a large pot of water to a boil. Add the green beans and boil for about 4 minutes, until crisp-tender. Drain and immediately transfer to a large bowl of ice water to stop the cooking process. Let cool for 5 minutes, then drain well in a colander. Once dried, transfer the green beans to the prepared baking dish.
Meanwhile, melt 1 tablespoon butter in a medium saucepan over medium heat. Add the mushrooms and onion and sauté for 4-6 minutes, stirring occasionally, until softened.
Add 3 more tablespoons of butter and stir until melted. Stir in the minced garlic and flour, cooking for 30 seconds.
Gradually add the chicken broth, milk, salt, and pepper, stirring until the mixture thickens, about 2-3 minutes. Remove from heat and pour over the green beans, ensuring they are evenly coated.
Melt the remaining 1 tablespoon butter and mix it with the breadcrumbs. Stir in the Parmesan cheese. Sprinkle this mixture evenly over the casserole.
Bake in the center of the oven for 35-40 minutes, until the filling is hot and bubbly. Check at 25 minutes—if the topping is browning too quickly, cover the dish loosely with foil.
Let the casserole rest for 5 minutes before serving.