Line a baking sheet with parchment paper. Preheat oven to 350° F.
Place softened butter in the bowl of your mixer or a large mixing bowl. Beat until light and fluffy, about 2 minutes.
Add brown sugar and beat until creamy. Add egg and beat well.
Stir in all remaining ingredients and mix until thoroughly incorporated.
Using a cookie scoop or a soup spoon, drop dough on to the prepared baking sheet.
Bake for 10 minutes, and then using the back of a flat spatula, press down on the cookies to make them a little bit flatter. Continue to bake for 1-2 more minutes, or until done. Remove from oven and cool completely before frosting.
Set aside about 1/3 to 1/2 cup of the frosting for making carrot decorations. Spread the remaining frosting evenly over the top of each cookie. Then dust some of the cookies with cinnamon or allspice, sprinkle some of them with chopped candied pecans and the rest decorate with a little frosting carrot on top. Or, you can just choose one of these topping ideas.
To make the carrot decorations, dye some of the remaining frosting orange and some green. Then put the frosting into two separate pastry bags, fitted with a leaf tip or a round tip and couplers to make a cute little carrot on top.
Store frosted cookies in a single layer in an airtight container in the refrigerator but remove to warm up about 10 minutes before serving.