Combine the cream and sugar in a heavy small saucepan. Add the cinnamon stick, and bring to a boil. Remove from heat, and let sit for 1 hour. Slowly whisk the yolks into the cream mixture. Add the rum. Stir over medium-low heat using a wooden spatula until the sauce has thickened enough to leave a path when a finger is drawn across the spatula. Strain the sauce through a fine sieve into a bowl. Discard the cinnamon stick. Add an ice cube to the mixture and cool to room temperature, then cover and refrigerate until chilled, 4 to 6 hours. This can be prepared 1 day in advance.
Heat oven to 350 degrees F. Line 2 large baking sheets with parchment paper. Trace 4 (4-inch) circles onto each parchment paper. Invert the parchment paper. Brush inside the circles lightly with the melted butter and sprinkle each circle with about 1 tablespoon sugar.
Remove filo from package and unroll it. Remove 1 sheet, and cover the remaining filo with plastic wrap and a damp towel. Brush the sheet of filo with melted butter, and sprinkle with 1 tablespoon sugar. Cut the filo in half crosswise. Turn 1 of the 1/2 filo sheets over (buttered and sugared side down), and crumple it until it conforms to the contours of one 4-inch circle on the parchment. Place on a circle. Crumple the second half of the filo sheet, and place on another circle in the same manner. Repeat the process with the remaining 3 filo sheets. Rewrap the remaining filo dough, and refrigerate or freeze for another use. Press the filo circles down gently to flatten them. Bake until golden, about 15 minutes. Transfer to a rack sugar side up. (These can be prepared ahead and set aside at room temperature.)
Melt the butter in a large heavy nonstick skillet over low heat. Add the sugar, and stir until dissolved. Add the apples, increase heat to medium high, and cook until just tender and caramelized, about 15 minutes. (This can be prepared ahead, cooled, covered, and set aside at room temperature.) To serve, place 1 round of pastry, sugar side down, on each plate. Overlap a quarter of the apples on top of the pastry. Top with another round of pastry, sugar side up. Ladle cinnamon custard around the pastry. Serve immediately.
(c)2002 Martha Stewart Living Omnimedia, Inc. All Rights Reserved