Preheat oven to 350 degrees. Line baking sheet with parchment paper or silicone baking mats and set aside.
Cream butter, coconut oil, sugar and molasses together until completely smooth. Scrape sides and mix again briefly. Stir in vanilla and egg. Then stir in dry ingredients until just combined. Scrape sides and stir again briefly. Stir in carrot and nuts by hand.
Scoop onto prepared baking sheets and bake 8-9 minutes or until edges are brown and cookies have poofed up slightly in the center. Cool 2 minutes on baking sheet, then transfer to cooling rack. Cool to room temperature.
For the frosting, place all ingredients into a small bowl and whip until smooth.
Spread each cookie generously with frosting and top with chopped pecans if desired.
*If you do not have coconut oil, simply use butter.**If you do not have granulated sugar and molasses, simply use brown sugar.