Preheat the oven to 375 degrees F (190 degrees C). Grease a 24-cup mini muffin tin or a 12-cup standard muffin tin.

Beat butter and brown sugar together in a mixing bowl until smooth. Stir in bananas, baby food, carrots, and eggs. Stir in flour, oat bran, baking soda, pumpkin pie spice, and salt until just combined. Spoon batter equally into the prepared muffin cups.

Bake in the preheated oven until a toothpick inserted into centers comes out clean, 15 to 20 minutes. Cool in the tins for 10 minutes before removing to a wire rack to cool completely. Store at room temperature for up to 2 days, or freeze.