Make the Onion Straws: Combine all the ingredients in a small bowl, toss to mix, and arrange in a single layer on a baking sheet. Set aside.
Make the Casserole: Bring a small pot of water to boil and season lightly with salt. Boil the green beans until tender, but still crispy, about 5 minutes. Drain and chill beans in an ice water bath or under running cold water. Drain again and transfer to a bowl.
Heat the oil in a large pan over medium-high heat. Add onion, mushrooms, salt, garlic, and pepper and cook, stirring, about 2 minutes. Add the chicken broth and thyme and cook until almost all the liquid is evaporated, about 3 to 4 minutes. Transfer the mushroom mixture to the bowl of green beans, add the sour cream, and toss to combine.
Pour the green bean mixture into the glass pie pan. Bake the casserole and onion straws on separate racks until the straws are well browned and crispy (almost burnt, otherwise they will be soggy), about 15 to 17 minutes. Top the casserole with the straws, and serve garnished with thyme sprigs.