Heat the oven to 400 degrees F. Chop the applesauce apples and put them in a saucepan with 2 tablespoons of the brown sugar, the nutmeg, and 1/4-cup/60 ml water. Cook, stirring occasionally, until very soft. Put the apple mixture through a food mill to remove the skins and seeds. You should have about 1-1/2 cups/375 ml applesauce.
Peel and very thinly slice the royal gala apples. Toss with the remaining brown sugar and melted butter. Spread the applesauce in the base of the baked tart shell and arrange the sliced apples over top. Brush the apple with any butter and sugar remaining in the bottom of the bowl. Bake until the apples slices are very soft and golden, about 20 minutes, depending on their thickness. Remove the tart from the oven and let cool slightly. Unmold. Serve warm or completely cool for a firmer tart with vanilla ice cream or a spoonful of creme fraiche.