Preheat oven to 350 degrees.
Spread bread slices thinly on 1 side with 1/2 stick butter. In a small bowl stir together 1/4 cup granulated sugar, cinnamon, and nutmeg and sprinkle evenly over buttered sides of bread. Arrange bread, buttered sides up, on baking sheets and toast in batches in middle of oven until just golden, about 15 minutes. Cool cinnamon toast on racks and quarter each slice into triangles, reserving 24 triangles.
Peel, quarter, and core apples. Cut apples lengthwise into thin slices and in a bowl toss with lemon juice. In a large heavy skillet melt remaining 1/2 stick butter with brown sugar over moderately high heat, stirring, and add apples and water. Cook mixture, covered, over moderate heat, stirring occasionally, 10 minutes. Remove lid and cook apples until just tender and most liquid is evaporated, about 5 minutes more.
Butter a 9-cup oval gratin dish or a 13 by 9 by 2-inch shallow baking dish (about 3 quarts). Arrange 1 layer of cinnamon toasts in dish, breaking triangles to fit. Spoon half of apple mixture evenly over toasts and sprinkle evenly with 1/2 cup Cheddar. Top with another layer of cinnamon toasts and remaining apple mixture. Arrange reserved cinnamon toast, sugared sides up, overlapping slightly, in a ring around edge of dish. In a bowl whisk together eggs, remaining 1/2 cup granulated sugar, milk, and salt and pour slowly and evenly over apples. Chill pudding, covered, at least 1 hour and up to 1 day.
Preheat oven to degrees.
Bake pudding in middle of oven 30 minutes. Sprinkle remaining 1/2 cup cheddar evenly over top and bake pudding 15 minutes more.