Cover the bottom of a 9x9 cake pan in tin foil and grease the tin foil.
Preheat the oven to 350F.
Whisk together the brown sugar, vanilla extract, and melted butter in a large bowl. Add the egg in and whisk thoroughly.
In a smaller bowl, sift together the spices, salt, and flour.
Grate the carrot finely and whisk it into the wet ingredients.
In a measuring cup, stir the oats with the honey and add them to the wet ingredients, again stirring well.
Fold the dry ingredients into the wet ingredients and stir until combined.
Pour batter into the prepared pan and bake for 26 minutes.
Using a pastry blender or fork, combine the sugar, cream cheese, and butter until completely blended. Be sure to incorporate the butter entirely. Add milk as needed, in teaspoons, to get the desired consistency.
When the bars are cooled, spread the frosting over the entirety in and even layer, or alternatively, pipe out in a decorative fashion.
In the refrigerator: This will keep with the frosting in a sealed container for one week.
In the freezer: These will keep for 6 months, well wrapped, without frosting.