Position a rack in the center of the oven and preheat the oven to 350°F. Lightly butter a 13 x 9 x 2-inch baking dish.

To prepare the casserole: Bring a large saucepan of salted water to a boil over high heat. Add the green beans and cook just until they turn a brighter shade of green, about 2 minutes. Drain, rinse under cold running water, and drain well. Pat them dry with a clean kitchen towel. (To prepare the

green beans ahead, wrap them in paper towels after drying, transfer to a 1-gallon plastic zip-tight bag, and refrigerate for up to 1 day.)

Melt the butter in large skillet over medium-high heat. Add the mushrooms and cook, stirring occasionally, until they begin to brown, about 10 minutes. Stir in the shallots and cook, stirring often, until they soften, about 2 minutes. Sprinkle with the flour and stir well. Stir in the milk, broth, and Maggi and bring them to a boil, stirring often. (The mushroom sauce can be cooled, covered, and refrigerated for 1 day. Reheat in a large skillet over low heat.)

Stir the green beans into the mushroom sauce. Transfer them to the baking dish. Season with pepper. Bake until the sauce is bubbling, about 20 minutes. Remove the casserole from the oven and tent with aluminum foil to keep warm. Reduce the oven temperature to 200°F.

To deep-fry the shallots: Meanwhile, pour enough of the oil into a large, deep skillet to come halfway up the sides and heat until the oil is shimmering and reads 365°F on an instant-read thermometer. Place a wire cake rack in a rimmed baking sheet.

Mix the flour, salt, and pepper together in a brown paper bag. Mix the shallot rings and buttermilk in a medium bowl. In batches, remove the shallots from the buttermilk, letting the excess buttermilk drain back into the bowl. Add the shallots to the flour mixture and shake to coat. Transfer them to the oil and deep-fry until golden brown, about 1 1/2 minutes. Using a wire spider or a slotted spoon, transfer the shallot rings to the cake rack to drain and place them in the oven-to keep warm. Top the casserole with the fried shallots and serve.