Preheat oven to 350 degrees F. Line baking sheets with parchment paper.
In a large bowl, stir together oats, flour, flaxseed, cinnamon, nutmeg, baking soda and salt.
In a medium bowl or liquid measuring cup, whisk together applesauce, honey (or maple syrup), egg and vanilla. Mix in the melted butter.
Pour the wet ingredients into the dry and stir until just combined. Gently fold in the grated carrot and apple.
Use a large scoop or measuring cup to drop 1/4-cup portions of dough onto the prepared baking sheets. Use your fingers to gently shape the cookies and flatten slightly, as they won't flatten on their own during baking.
Bake 14-15 minutes, until set and lightly golden. (If baking more than one pan at a time, be sure to rotate the pans halfway through the baking time.)
Let cookies cool on the baking sheet for a few minutes before removing to a wire rack to cool completely.