Preheat the oven to 375 degrees F. Lightly mist a 2-quart baking dish with cooking spray. Toss the shallots with the oil and salt and pepper on a baking sheet, spread in a single layer, and bake until golden and crispy, stirring 3 to 4 times, about 20 minutes. Remove from the oven and immediately sprinkle the breadcrumbs over the shallots and toss well. Set aside to cool.
Meanwhile, in a small pot combine the milk, thyme, bay leaves and peppercorns and bring to a simmer. Turn the heat off and let steep for 20 minutes. Bring a large pot of salted water to a boil. Add the beans and cook until bright green and crisp tender, about 6 to 8 minutes. Drain well and run briefly under cold water. Drain again and set aside.
Strain the milk into a large skillet, add the mushrooms and season with salt and pepper. Bring to a simmer over medium heat and cook until the mushrooms have softened, 10 to 15 minutes. Stir the broth and flour together and pour into the skillet; cook, stirring, until the sauce thickens, 6 to 8 minutes. Remove from the heat and add the beans and nutmeg. Toss until the beans are well coated in the sauce and transfer to the baking dish. Scatter with the shallots and bake until the sauce is bubbly and the shallots are crisp, about 20 minutes. Let stand 10 minutes before serving.
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