Preheat the oven to 375°F (190°C). Butter a 9x13-inch baking dish.
Blanch the green beans: bring a large pot of salted water to a boil. Add the green beans and cook for 3 to 4 minutes, until bright green and crisp-tender. Drain and immediately transfer to a bowl of ice water to stop the cooking. Drain well and pat dry.
Sauté the mushrooms: in a large skillet over medium-high heat, melt 2 tablespoons butter with the olive oil. Add the mushrooms in a single layer and cook, without stirring, for 4 to 5 minutes until deeply browned. Stir and cook 3 minutes more. Add the onion and cook 3 minutes until softened, then add the garlic and cook 30 seconds until fragrant.
Make the sauce: sprinkle the flour over the mushrooms and stir for 1 minute to cook off the raw flour taste. Slowly whisk in the milk, cream, and broth until smooth. Bring to a gentle simmer and cook 4 to 5 minutes, stirring often, until thick enough to coat the back of a spoon. Stir in the soy sauce, Dijon, nutmeg, salt, and pepper. Taste and adjust the seasoning — the sauce should taste rich and well-salted on its own.
Assemble: in a large bowl, toss the blanched green beans with the mushroom sauce and 1 cup of the crispy fried shallots. Spread into the prepared baking dish in an even layer.
Make the topping: in a small bowl, stir together the panko, Parmesan, and the remaining 2 tablespoons melted butter until the crumbs are evenly moistened.
Bake: sprinkle the panko mixture and the remaining 1/2 cup crispy fried shallots over the casserole. Bake for 25 to 30 minutes, until the sauce is bubbling around the edges and the topping is deep golden brown.
Rest and serve: let the casserole rest for 10 minutes so the sauce thickens slightly. Sprinkle with fresh parsley and serve warm.