Arrange the fillets in a shallow dish and pour the milk over them. Let soak for at least 5 minutes and up to 20 minutes. Set up your work area so that you can move quickly: position your serving plates or platter, the milk soaking pan, a pile of paper towels, and the seasoned flour on a plate. Have the melted butter in a small, heavy-based saucepan, and the lemon juice and parsley ready for action.
In one or two large frying pans, heat the clarified butter over medium-high heat until very hot but not quite smoking. Lift a fillet from the milk, blot it on the paper towels, dip it into the flour, and shake off the excess. Carefully lay the fish in the hot fat. Repeat with the other fillets, but don't overcrowd the pan or you'll have trouble flipping. Adjust the temperature to keep the fat sizzling briskly but not burning. Cook the fish until golden on one side, about 2 minutes. With a slotted spatula, a large spoon, and great care, gently flip the fish.
When the second side of the fish is golden brown and the flesh is tender when poked with a sharp knife in the thickest part, use the slotted spatula to remove the sole, set it on paper towels to drain briefly, and arrange on the warm platter or plates.
When all the fillets are cooked, heat the melted whole butter carefully over medium-high heat, swirling the pan occasionally, until the butter is fragrant and the milk solids turn nutty brown; remove the pan from the heat so the butter doesn't keep cooking, but keep it hot.
Working fast, pour 1 Tbs. of the lemon juice evenly over each fish and sprinkle on the parsley. Pour about 1-1/2 Tbs. of the hot browned butter on each fish -- if all has gone well, you'll see and hear a delicious sizzle. (If there's no sizzle, it will still taste great.) Serve immediately.