• Preheat the oven to 375 degrees F.

To make the topping

• Heat a medium-sized skillet over medium heat. Add the shallots and sauté for 3-5 minutes, or until they are softened.
• Reduce the heat to medium-low and add the bread crumbs, cheese, and thyme. Sauté for 5-6 additional minutes, stirring frequently to avoid burning. Remove from the heat and set aside.

To prepare the green beans

• Boil a large pot of water. Add the green beans and blanche for 2-4 minutes. Drain them into a colander and set aside.

To make the sauce

• Heat the olive oil in a large sauté pan over medium-high heat. Add the remaining ⅓ cup of shallots and sauté for a couple of minutes. Add the mushrooms, stirring to coat them with the oil. Season with salt and pepper. Sauté for 6-8 minutes, stirring occasionally.
• Sprinkle the flour over the mushrooms, stirring to coat. Continue stirring for about a minute. Slowly add the broth and milk. Bring it to a low boil, and cook for about 3 more minutes, or until it starts to thicken up, stirring occasionally. Stir in the rest of the grated cheese.

To assemble the dish

• Add the blanched green beans and mix well. Taste, and season with salt and pepper, if needed. Pour everything into the prepared baking dish.
• Top the beans and sauce with the toasted bread crumbs and bake for 30 minutes.