Peel and core apples. Slice apples into thin concentric circles making sure to keep the shape of the apple. Combine sugar and water in saucepan. Stir with whisk to combine, remove whisk and do not stir anymore with the whisk. Cook sugar and water over medium high heat, continuing to brush down sides of pan with wet pastry brush to avoid sugar from crystalizing. Cook caramel to deep amber color. Remove from heat and butter with a clean whisk, piece by piece, to make a thick caramel. Pour into 8-3 inch high large muffin pans or an 8-inch cake pan with 3-inch sides.
Add thinly sliced apples to caramel, pressing down into caramel. Completely fill to brim with as many apples as possible. Bake in oven for total of 1½ hours, removing from oven to pat down apples every 30 minutes. Allow to cool 30 minutes for small individual tartes or 1 hour for 1 large tarte.
Roll puffy pastry or pie dough to little larger circles than tartes. Poke with fork. If using puff pastry then cover and bake with a rack to avoid puff from rising too much. Bake until lightly golden brown. Place over cooled apple tartes and invert onto cooked pastry rounds. Serve at room temperature or gently reheat at 350 for about 10 minutes.
In a mixer, whisk cream on medium speed to soft peaks. Add vanilla and rum. Add part of sugar. Continue whisking to form medium peaks. Add rest of sugar and whisk to stiff peaks.
Serve tartes and rum crème Chantilly together.