Preheat oven to 375°F. Grease a muffin pan or line with paper muffin liners.
In a large mixing bowl, use an electric mixer to beat the brown sugar, applesauce and butter together until fluffy. Add the bananas, milk, eggs and vanilla; mix well. In a separate bowl, whisk together the remaining (dry) ingredients. Stir the dry ingredients into the wet mixture just until moistened. Divide the batter between the prepared muffin cups.
Bake for 15 to 20 minutes, or until a toothpick inserted into the center of a muffin comes out clean. Cool in the cups for 5 minutes, then remove the muffins and place on a wire rack to cool completely.