Preheat oven to 375 degrees. Coat 2 mini-muffin tins with cooking spray. Place applesauce and dates in a medium saucepan. Cook over medium heat, stirring frequently, until mixture is reduced to 1 1/4 cups, 15 to 20 minutes. Spread in an even layer on a rimmed baking sheet, and let cool completely.
Transfer to a large bowl, and stir in bran, buttermilk, egg, honey, ginger, and vanilla. Let stand for 10 minutes.
Meanwhile, whisk together flour, flaxseed, baking soda, salt, allspice, and 1/4 cup oats. Stir into bran mixture.
Spoon batter into prepared tins, filling to the brims. Sprinkle remaining 1 tablespoon oats over muffins. Bake until a toothpick inserted into center of 1 comes out clean, 21 to 23 minutes. Let cool completely in pans on wire racks.
Muffins can be stored in an airtight container at room temperature for up to 3 days.