Adjust an oven rack to middle position and preheat oven to 425 degrees. In a Dutch oven or stock pot, bring 4 quarts water and 2 tablespoons salt to boil. Fill a large bowl with ice water.
Add green beans and cook until bright green and tender-crisp, about 6 minutes. Drain in colander and immediately transfer to ice water. Spread on paper towels to dry.
To the same pot, melt the butter over medium-high heat until foaming. Add mushrooms, garlic, 3/4 teaspoon salt, and 1/8 teaspoon pepper. Cook until the mushrooms have released most of their liquid, about 5 minutes.
Stir in flour and cook for one minute. Whisk in broth, scraping up any brown bits from the bottom of the pan. Bring to a boil, whisking constantly.
Reduce heat to medium. Whisk in cream and simmer until the sauce has thickened and reduced to 3 1/2 cups, about 10 to 15 minutes. Season to taste with salt and pepper.
Add the green beans to the pot with the mushroom sauce and toss until evenly coated. Pour into a 3-quart or 9-inch by 13-inch baking dish. Sprinkle with topping.
Bake until the top is golden brown and the sauce is bubbling around the edges, about 15 minutes. Serve immediately.