Discard the small, tough muscle from the side of each scallop and in a bowl let the scallops soak in the milk, covered and chilled, for 1 hour. While the scallops are soaking, in a large saucepan steam the spinach in the water clinging to its leaves, covered, over moderately high heat for 3 to 4 minutes, or until it is wilted. Drain the spinach in a colander, squeeze it dry in a kitchen towel, and coarsely chop it. Drain the scallops and pat them dry. In a small skillet cook the spinach in 2 tablespoons of the butter over moderate heat, stirring, until it is hot and season it with the nutmeg and salt and pepper to taste. Spread the spinach on a heated platter and keep it warm, covered.
Divide the oil and four tablespoons of the remaining butter between two heavy skillets and heat the fat over moderately high heat until it is golden. While the fat is heating, working in batches, dredge the scallops in the flour and in a large sieve shake off the excess. Divide the scallops between the skillets, in the hot fat saute them, stirring and separating them with a fork, for 1 to 2 minutes, or until they are opaque and light golden, and transfer them with a slotted spoon to the bed of spinach. Season the scallops with the lemon juice and salt and pepper and sprinkle them with the parsley.
In a small saucepan heat the remaining five tablespoons butter over moderately high heat until it is golden brown and has a nutlike fragrance and pour the brown butter carefully over the scallops.