Make the crispy onions first: toss the thin-sliced onions with the flour, salt and pepper until every strand is coated, shaking off the excess. Heat the oil in a large skillet over medium-high heat until it shimmers. Fry the onions in two batches, stirring now and then, 5–7 minutes per batch, until deep golden and crisp. Lift them out with a slotted spoon onto a paper towel-lined plate and sprinkle lightly with salt. Set aside.

Heat the oven to 375°F (190°C). Grease a 9x13-inch baking dish (or a 2 1/2-quart casserole) with butter.

Blanch the beans: bring a large pot of water to a rolling boil, add the 1 tablespoon salt, then the green beans. Cook 4–5 minutes, until bright green and just tender-crisp. Drain and plunge straight into a big bowl of ice water to stop the cooking. Drain again and pat dry.

Make the sauce: melt the butter in a large skillet over medium-high heat. Add the mushrooms and cook, without stirring too much, 6–8 minutes, until they give up their liquid and turn golden brown. Stir in the garlic and thyme and cook 30 seconds, until fragrant.

Sprinkle the flour over the mushrooms and stir for 1 minute. Slowly pour in the broth, stirring as you go, then stir in the milk and cream. Simmer gently 4–5 minutes, stirring often, until the sauce thickens enough to coat the back of a spoon. Stir in the Worcestershire sauce, nutmeg, salt and pepper. Taste and adjust the salt.

Fold the drained green beans into the sauce until every bean is coated, then scrape it all into the prepared dish and smooth the top.

Bake uncovered 20–25 minutes, until bubbling hot all the way through and at the edges.

Scatter the crispy fried onions thickly over the top and return to the oven 3–5 minutes, just until the onions are warmed and extra crisp. Rest 5 minutes before serving. Serve warm, straight from the dish.