Butter a round casserole and line with parchment paper. The parchment should reach up, well above the casserole.
Core apples, slice in half, and peel. Store peeled apples in lemon water. Slice the apples with the 1mm food processor disc, placing them core down.
Rub the sugar cubes all around the oranges, absorbing the orange oil. Crush the cubes and set aside.
Sprinkle half the sugar mixture on the bottom of the casserole. Arrange the first layer of apples on the bottom of the casserole. This will be the visible top of the cake, so arrange them in a pleasing pattern. Layer about half of the remaining apples on top, then sprinkle on the remaining sugar, and finish with the remaining apples.
Cut a circle out of parchment, and place on top of the casserole (and apples) and place a plate as a weight on top of the parchment. Bake in the oven at 175 (yes that's right, 175) for 12 - 13 hours. Remove from the oven, and allow to sit for 5 minutes. Pull the parchment from the sides and allow to sit another 5 minutes. Invert the cake onto the serving plate.
Mix equal amounts of the marscapone and the yogurt and use as a garnish. Pour some carmel sauce over the cake.
Heat oven to 175 degrees F.
Generously butter a 1 1/2-quart-tall souffle dish and a wide strip of parchment paper to form a collar extending at least 3 inches above the rim of the dish. Press the collar, buttered side inward, against the inside of the dish. Chill until butter is set and the paper sticks to the dish.
To extract the zest from the skins of the oranges, rub them with the sugar cubes so that the cubes soften and turn bright orange. Wrap the sugar cubes in plastic wrap, and crush them with a rolling pin.
Set an apple 1/2, cut side down, on a board and cut it crosswise into the thinnest possible slices, about 1/8-inch thick. Alternately, slice apple halves using a mandoline. Sprinkle bottom of souffle dish with sugar. Arrange a layer of apple slices in a floral pattern in the bottom of the souffle dish. Top this first layer of apple with more apples slices arranged across the others like ripples in a pond. (This crossed pattern of slices ensures that the cake holds together when unmolded.) Sprinkle the second layer with the crushed orange sugar. Continue filling the souffle dish until the apples, held in place by the paper collar, extend at least 3 inches above the rim. (They will shrink down into the souffle dish during baking.) Cover them with a round of parchment paper. Top with a small stack of plates slightly smaller than the souffle dish to seal and weigh down apples.
Bake until apples are much reduced and meltingly soft when pierced with a skewer, 12 to 14 hours. Tear off the top of the paper collar, and let the cake cool to room temperature.
Combine mascarpone and yogurt in a small bowl. Unmold cake onto a warm platter. The top should be lightly caramelized with a little syrupy juice running down the sides. Serve warm, with caramel sauce and mascarpone-yogurt combination.