Heat oven to 400° F. Coat a shallow 2 1/2- to 3-quart baking dish with oil; set aside.
Heat the oil in a large skillet over medium-high heat. Add the shallots, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Cook, stirring occasionally, until soft, 6 to 8 minutes. Add the wine and simmer until evaporated, 4 to 6 minutes.
Meanwhile, in a large bowl, whisk together the eggs, cream, milk, nutmeg, 1/2 teaspoon salt, and 1⁄4 teaspoon pepper.
Squeeze the spinach to remove excess liquid. Stir the spinach, shallots, Gruyère, and Parmesan into the egg mixture. Transfer to the prepared baking dish.
Bake until bubbling and the top is golden brown, 45 to 55 minutes.
To Bring It: If the casserole is warm, transport it in an insulated bag or a cardboard box lined with towels. Or you can make it a day in advance and, upon arrival, reheat it, covered, at 350° to 400° F for 45 to 50 minutes.