Blanch green beans in boiling salted water for 4–5 minutes, then transfer immediately to an ice bath; drain well.
Heat butter and olive oil in a skillet. Sauté diced onion until softened. Add garlic and cook briefly until fragrant.
Add mushrooms and sauté until browned and reduced, allowing moisture to evaporate fully.
Sprinkle flour over the mixture and cook 1–2 minutes to form a roux.
Slowly whisk in chicken broth and then heavy cream, stirring until smooth and slightly thickened.
Season with salt, black pepper, and smoked paprika (if using).
Fold in the blanched green beans and grated Parmesan; adjust seasoning to taste.
Transfer mixture to a baking dish. Top evenly with crispy fried onions.
Bake at 375°F (190°C) for 20–25 minutes, until hot and bubbling with a golden, crisp topping.