1. Preheat oven to 425 degrees. In a large skillet, heat oil over medium. Line a baking sheet with paper towels. In a large bowl, toss together shallots and 1 1/4 cups flour until shallots are evenly coated. In batches, shake off excess flour from shallots and fry until golden and crisp, about 5 minutes, adjusting heat if shallots are browning too quickly. With a slotted spoon, transfer to sheet and season with salt. Set aside. (Store in an airtight container at room temperature with a layer of paper towels underneath, up to 2 days.)

2. In a large pot of boiling salted water, cook green beans until crisp-tender, 6 minutes. Drain and rinse with cold water to stop the cooking. Pat dry. (To store, refrigerate in a resealable plastic bag, up to 1 day.)

3. In a large saucepan, melt butter over medium-high. Add mushrooms and cook until liquid has evaporated, about 7 minutes. Season with salt and pepper. Add 1/4 cup flour and cook, stirring, until incorporated, about 1 minute. Whisking constantly, gradually add broth, then milk. Bring to a simmer. Cook, stirring occasionally, until sauce thickens, about 5 minutes. Season with salt and pepper. (To store, refrigerate with plastic wrap on surface, up to 2 days. Reheat before using.)

4. Add green beans to mushroom sauce and toss to coat. Transfer mixture to a 9-by-13-inch baking dish. Bake until bubbling around edges, about 15 minutes. Serve topped with fried shallots.