Bake these cakes in ramekins, and eat them from the dish with a spoon.
Preheat oven to 400°.
Combine first 4 ingredients in a small saucepan; place over high heat. Bring to a boil; reduce heat, and simmer 3 minutes or until apricots are plump. Remove from heat; cool.
Lightly spoon flour into dry measuring cups; level with a knife. Combine flour, bran, baking soda, and salt in a medium bowl; stir well with a whisk. Make a well in center of mixture.
Combine brown sugar and next 4 ingredients (through egg white) in a small bowl; stir well with a whisk. Add sugar mixture to flour mixture, stirring just until combined. Gently fold in apricot mixture.
Spoon 1/2 cup batter into each of 7 (6-ounce) ramekins coated with cooking spray. Place ramekins on a jelly-roll pan. Bake at 400° for 15 minutes or until golden brown. Place on a wire rack. Serve warm or at room temperature.