Preheat oven to 350 degrees F. Line a baking sheet with parchment paper.
Pour rum over raisins and microwave on high for 25 seconds to plump raisins (let sit 10 minutes).
In a large bowl, combine the apples, lemon juice, cinnamon, sugar, and plumped raisins.
In a medium bowl, whisk together the oil and egg whites, being sure to mix them well.
Place 1 sheet of phyllo on work surface and lightly brush with the oil-egg white mixture. Sprinkle 1 tablespoon each of sugar, panko, and almonds, then top with another layer of phyllo and repeat the process four times. Place the last sheet of phyllo on top of the sugar mixture and brush it with the with oil-egg white mixture, but do not top with more sugar, panko, and almonds.
Place the apple mixture on the lower 1/3 of the phyllo stack, being sure to leave a 2-inch border. Gently lift the bottom edge of the phyllo stack to cover the filling, then fold the side edges over. Continue to roll the stack away from you, until the filling is completely sealed in and the seam is on the bottom.
Transfer to the lined baking sheet, brush the top with the oil-egg white mixture and sprinkle with 1 more tablespoon of sugar. Place in oven and bake for 30 minutes. Let cool before serving.
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