Heat the oven: Preheat to 375°F (190°C). Lightly grease a 9x13-inch baking dish.
Blanch the beans: Bring a large pot of salted water to a boil.
Add green beans and cook 3–4 minutes until bright green and crisp-tender. Drain and immediately transfer to a bowl of ice water to stop cooking. Drain again and set aside.
Sauté the mushrooms: In a large skillet, heat olive oil and butter over medium-high.
Add mushrooms and a pinch of salt. Cook, stirring occasionally, until they release moisture and turn golden, about 6–8 minutes.
Add aromatics: Lower heat to medium. Stir in shallot, garlic, and thyme.
Cook 1–2 minutes until fragrant.
Build the sauce: Sprinkle flour over the mushroom mixture and stir for 1 minute to cook off the raw taste. Slowly whisk in broth, scraping up any brown bits. Add milk and simmer 3–5 minutes, stirring, until slightly thickened.
Finish the creaminess: Remove from heat.
Whisk in Greek yogurt and Dijon until smooth. Season with salt, pepper, and nutmeg. Taste and adjust.
Combine: Add the blanched green beans to the skillet and toss to coat.
Transfer to the prepared baking dish, spreading evenly.
Make the topping: In a small bowl, combine panko, almonds, and Parmesan. Add a light drizzle of olive oil or a few sprays to help it crisp, then toss.
Top and bake: Sprinkle the topping evenly over the casserole. Bake 18–22 minutes, until bubbling at the edges and the top is golden.
Rest and serve: Let it sit 5–10 minutes so the sauce sets slightly.
Serve warm.