Preheat the oven to 400°F.

Spray an 8- or 9-inch square baking dish with nonstick cooking spray. Set aside.

Cut the green beans into 1 1/2-inch pieces. Slice the mushrooms, finely chop the onion, and mince the garlic.

Bring a large pot of salted water to a boil. Once the water is boiling, add the green beans. Blanch until beans are bright green, about 3 minutes. Drain the green beans in a colander and rinse with cold water.

Set a large sauce pan over medium-high heat and add the butter. Once the butter is melted, add the onions and mushrooms to the pot. Sauté until the mushrooms are browned and shrunken and the onions are soft, 6-7 minutes. Add the garlic and thyme and cook until fragrant, 30 seconds longer.

Sprinkle the flour over the mushrooms and onions. Stir with a spatula to coat. Slowly pour the broth into the pot, whisking to dissolve the flour. Pour in the half and half, and season with salt and black pepper.

Cook the sauce, whisking constantly, until it begins to bubble and thicken. Reduce the heat to medium-low and allow the sauce to simmer, whisking occasionally, until it is reduced by about a third, 5-7 minutes.

Add the green beans to the pot and stir to coat with the sauce. Pour the green bean mixture into the casserole dish. Sprinkle the shredded cheeses in an even layer over the top.

Bake the green bean casserole on middle rack of oven until it is bubbling at the edges and the cheese is golden brown, 20-25 minutes.

Check to see that casserole is done. Remove from oven or add time as needed.

Allow casserole to rest for 10-15 minutes before serving.