Bring a large pot of water to a boil; add a big pinch of salt and the green beans. Cook for about 5 minutes, the beans should still be crisp, they will be cooked more in the oven. Drain them and set aside. Butter a baking dish large enough to hold the green beans with 1 tablespoon butter and set aside.
Heat the oven to 375 degrees F.
Tear the bread into 2-inch pieces, put them into a bowl, and add 1 tablespoon chives, 1 tablespoon thyme, 1 tablespoon rosemary, 2 tablespoons Parmesan, and 3 tablespoons olive oil. Stir well to coat and spread onto a baking sheet. Bake just until the bread just starts to turn golden, about 10 minutes. Remove from the oven and set aside.
Meanwhile, melt the remaining butter and olive oil in a large skillet over medium heat. Add the mushrooms and shallots, season with salt and pepper, and cook until the mushrooms have released their liquid, about 10 minutes. Pour in the heavy cream, add the remaining thyme and chives, and cook for another 5 minutes. Add the green beans and stir well. Put the green bean mixture into the prepared baking dish, top with the croutons, and sprinkle on the remaining Parmesan. Bake until everything is hot and bubbling, about 20 to 25 minutes.