For the pancake batter: Sift the flour, sugar, baking powder, and salt into a small mixing bowl and make a well in the center. Whisk the milk, egg, melted butter, and vanilla until smooth in a 1-quart measure. Pour into the dry ingredients, and whisk until creamy. Cover and let stand while you prepare the apples.

For the apple mixture: Quarter each apple, then peel, core, and slice each quarter crosswise 1/8-inch thick, letting the slices fall into a large mixing bowl. Add the sugar mixture and lemon juice and toss well. Melt the butter in a heavy 12-inch skillet over moderate heat, then let it foam up and subside. Add the apple mixture and saute 2 minutes, stirring often. Pour in the water, reduce the heat to low, cover, and cook 5 minutes. Uncover, raise the heat to moderate and boil, uncovered, shaking the skillet often, for 1 1/2 to 2 minutes, just until all juices boil away. Scoop the skillet mixture into a 1-quart measure and reserve.

Preheat the oven to 150 to 200 degrees F.

To cook the pancakes: Generously oil the bottom and sides of a well-seasoned 10-inch omelet pan or spray with nonstick vegetable cooking spray and set over moderate heat for 1 minute. Remove the pan from the heat, pour in a scant 1/3 cup of the pancake batter, and tilt the pan first to 1 side, then to another, until the batter coats the bottom of the pan in a thin, even layer. Set the pan on a hot pad on the counter, then by hand, arrange 1/2 cup of the apple slices on top of the batter in the pan, distributing them as evenly as possible. Pour in another scant 1/3 cup batter, covering the apples as uniformly as possible. Tilt the pan gently to distribute the batter more evenly, if necessary.

Set the skillet over moderate heat and cook the pancake, uncovered for 1 minute. Reduce the heat to low and cook, uncovered, 2 minutes longer, until the pancake has dried around the edge and holes begin to appear on top. Spray a small, thin-bladed spatula with the cooking spray, carefully loosen the pancake around the edge, then shake the pan over the heat several times until the pancake moves freely. Quickly spray a large, flat round plate with the cooking spray and ease the pancake onto it right side up. Using potholders to protect your fingers, invert the omelet pan on the plate, then invert once again so the pancake is in the pan uncooked side down. Set over moderate heat and cook the pancake, uncovered, for 2 minutes. Slide onto a large round, ovenproof plate, cover with foil, and set in the warm oven. Cook the remaining pancakes the same way, recoating the skillet with cooking spray before each new pancake. As each new pancake finishes cooking, slide on top of the foil covered pancake, top with more foil and return to the warm oven. When ready to serve, slide each pancake onto a heated plate and dust with vanilla sugar.

Broiler method: Set the broiler rack 6 to 7 inches below the heating element and preheat the broiler. Follow the basic recipe, but be sure that the omelet pan or skillet you use has a flameproof handle. As soon as a pancake has browned 3 minutes in the skillet on top of the stove, transfer to the broiler and broil 3 to 3 1/2 minutes, until nicely tipped with brown. Carefully loosen the pancake around the edge with a spatula, ease right side up onto a large round plate, cover with foil, and keep warm. Cook the 3 remaining pancakes the same way. Dust with sugar and serve.