Preheat the oven to 400 degrees. Line 2 sheet pans with parchment paper. Unfold the sheets of puff pastry and cut each sheet into quarters. Place the pastry pieces on the prepared sheet pans and refrigerate while you prepare the apples. Peel and core the apples. Slice the apples in half (from the top of the apple), then slice crosswise in 1/4- inch-thick slices. (8 slices per ½ apple). Place 6 overlapping slices of apples diagonally across each piece of pastry and place a slice of apple on each side. Sprinkle the sugar over all the pastry pieces and dot with the butter. Bake for 40 minutes, until the pastry is browned and the edges of the apples start to brown. Rotate the pan once during cooking. If the pastry puffs up in one area, cut a little slit with a knife to let the air out. When the galettes are done, heat the apricot jelly together with the rum and brush the apples and pastry completely with the jelly mixture. Loosen the galettes with a metal spatula so they don’t stick to the paper. Serve warm or at room temperature with a scoop of ice cream, walnuts and Maple Balsamic Syrup over top. Make the Balsamic Syrup. In a deep saucepan over medium low heat, bring the balsamic vinegar and maple syrup to a boil and cook until it reaches 230 degrees. Set aside and allow the mixture to cool to room temperature. (When cooled, it should be the consistency of honey.) In a pot of boiling water, blanch the walnuts in a pot of boiling water for 2 minutes. Drain walnuts. Heat the frying oil to 300 degrees. Toss the walnuts in the confectioner’s sugar with a pinch of salt until well coated. Heat a skillet with 1 TBSP oil. Over medium high heat, fry the sugar coated walnuts until a deep golden brown, stirring often. Do not burn walnuts. Allow to cool on a silicone sheet. Place one scoop of ice cream on each apple galette. Sprinkle candied nuts over ice cream and drizzle with balsamic syrup. Serves 8.