1. Whisk together flours, bran, salt, and baking soda in a medium bowl; set aside. Put brown sugar and zest in the bowl of an electric mixer fitted with the paddle attachment; mix on medium speed until combined. Beat in butter until fluffy. Beat in egg until pale and smooth. Reduce speed to low. Mix in the flour mixture in 3 batches, alternating with the applesauce. Divide dough in half; wrap in plastic. Refrigerate until firm, about 2 hours.

2. Preheat oven to 375 degrees. Roll out 1 piece of dough between 2 lightly floured sheets of parchment to a 9 1/2-by-11 1/2-inch rectangle, about 1/8 inch thick. If dough becomes too soft, refrigerate until firm. Remove top piece of parchment. Trim dough 1/2 inch on all sides. Transfer rectangle on parchment to a baking sheet.

3. Cut rectangle in half lengthwise. Spread 1/4 cup jam down half of each length, leaving a 1/2-inch border. Fold dough over; pinch to seal. Refrigerate until firm, about 20 minutes. Repeat with remaining chilled dough and jam.

4. Cut crosswise into 1 1/2-inch-wide bars, but do not separate until after baking. Bake until golden brown, about 20 minutes. Let cool on sheet on a wire rack.