Pre-heat oven to 400. Line 12 muffin tins with parchment or silicone muffin liners. Spray with cooking spray if not using silicone liners.
Make your flax eggs and set aside.
In a large mixing bowl, stir together dry ingredients: flour, baking powder, baking soda, salt, cinnamon, coconut sugar, and brown sugar. Set aside.
In a separate medium bowl, whisk together the melted coconut oil (slightly cooled), applesauce, flax eggs, and vanilla. Stir the wet ingredients into dry ingredients until just incorporated. Gently stir in the shredded carrot and mashed banana until fully combined, but don't overmix.
Scoop batter into muffin tins, approximately 2/3rds full. Optional: sprinkle with chopped nuts before baking. Bake the muffins for 18-20 minutes until muffins bounce back when lightly touched and/or a toothpick comes out clean.
Let the muffins cool in the pan for 2-3 minutes before transferring to a wire rack. Serve once the muffins are cool enough to handle, yet still warm. Let any leftovers cool completely and store in an airtight container at room temperature for 1-2 days, or in the fridge.