Preheat oven to 350 degrees F. Lightly spray a baking dish with cooking spray.

Bring a large pot of water to a boil. Add green beans and boil for about 4 minutes, until crisp-tender. Transfer cooked green beans to a large bowl of ice water. Let cool in ice water for 5 minutes, then drain in a colander. Dry out the large bowl and place the dried green beans in it.

Meanwhile, melt 1 tablespoon of the butter in a medium skillet over medium-low heat. Add the mushrooms and onion and cook until soft, 4-6 minutes, stirring occasionally. Transfer mushrooms and onion to the bowl with the green beans.

In a small saucepan, melt 4 tablespoons of the butter over medium heat. Add the minced garlic and then whisk in the flour until smooth. Cook, whisking constantly, for 30 seconds. Whisk in the broth and half and half. Bring to a low simmer and continue to whisk constantly until thickened. Remove from the heat and stir in the salt, pepper, and Greek yogurt. Let the mixture cool slightly and then whisk in the egg.

Pour sauce into the bowl with the green beans and mushrooms. Stir gently to combine. Transfer to the prepared baking dish.

Melt the remaining tablespoon of butter and stir into the breadcrumbs. Stir in the grated cheddar. Sprinkle breadcrumb topping over the casserole.

Bake for 45 minutes, until filling is hot and bubbly. Check after 25 minutes and cover with foil if topping is browning too quickly. Let stand 10 minutes before serving.