Preheat the oven to 375 degrees F (190 degrees C). Grease a 12-cup muffin tin or line cups with paper liners.
Mix brown sugar and oil together in a bowl. Stir in applesauce and mashed banana. Mix in apple, zucchini, and carrots.
Combine whole wheat flour, all-purpose flour, flax seeds, baking powder, cinnamon, and salt in another bowl. Fold flour mixture into the sugar mixture. Pour batter into the prepared muffin cups; sprinkle chopped walnuts on top.
Bake in the preheated oven until a toothpick inserted into the center comes out clean, about 25 minutes.