Make filling: Bring dates and cider to a boil in a medium saucepan over medium-high heat. Reduce heat to medium, simmer until dates are soft and almost all liquid has reduced, about 10 minutes. Let cool completely. Transfer to a food processor. Puree date mixture until smooth, set aside.
Make dough: Whisk together flours, bran, salt, and baking soda in a medium bowl, set aside. Put sugar and zest in the bowl of an electric mixer fitted with the paddle attachment. Mix on medium speed 30 seconds. Add butter, mix until combined, about 1 minute. Add egg, mix until fluffy. Mix in flour in 3 batches, alternating with 2 batches of applesauce. Divide dough in half. Wrap each half in plastic; shape into a rectangle. Refrigerate until firm, about 2 hours.
Preheat oven to 375 degrees. Roll out 1 piece of dough between two lightly floured sheets of parchment paper to a 9 1/2-by-13-inch rectangle. Dust with more flour, if needed. Remove top piece of parchment. Trim dough to a 9-by-11-inch rectangle. Transfer rectangle on parchment to a baking sheet.
Cut rectangle in half lengthwise. Spread a heaping cup filling down half of each length, leaving a 1/4-inch border. Fold dough over; pinch to seal. Refrigerate until firm, about 20 minutes. Repeat with remaining chilled dough and filling.
Bake until golden brown, about 20 minutes. Let cool on sheets on wire racks 5 minutes. Cut into 1 1/2-inch bars. Transfer bars to racks using a spatula; let cool. Bars can be stored in airtight containers up to 3 days.
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