Preheat the oven to 350 degrees F. Line a cookie sheet with parchment paper or a silicone baking mat, then set aside. If needed, cook, mash, and cool the carrots: Peel and cut the carrots into large pieces (cut baby carrots in half), then place in a small saucepan. Cover with water, then bring to a boil over high. Let boil until the carrots are tender, about 6 minutes. Mash and let cool to room temperature (you can also place the carrots in the freezer to cool more quickly).
In a large mixing bowl, cream together the butter, shortening, and sugar until fluffy. Beat in the eggs one at a time, then the orange zest, orange juice, and vanilla extract. Stir in the mashed carrots.
In a separate bowl, stir together the all purpose flour, whole wheat flour, baking soda, and salt. Add to the wet ingredients, then mix gently, just until the flour disappears. Fold in the cranberries.
Drop by rounded teaspoons (the cookies are very small—do not make them larger or they will not bake as well) onto the prepared cookie sheets. Bake for 8 minutes, until the edges are barely golden. Let rest for 2-3 minutes on the baking sheet, then transfer to a wire rack to cool completely.
For the frosting: In a mixing bowl, beat together the powdered sugar, butter, orange juice, and vanilla extract. Frost the cooled cookies, then enjoy!
*My grandma's original recipe called for all shortening. If you prefer to stick to the original, use an additional 1/2 cup vegetable shortening in place of the butter. If you prefer to use ALL butter, you can swap it out for the shortening, but the cookies will have a heavier texture.
TO STORE: Store leftover cookies in an airtight container at room temperature for up to 5 days.
TO FREEZE: Place cookies in an airtight freezer-safe storage container in the freezer for up to 3 months. If freezing the cookies, I recommend waiting to frost them if possible (or freezing the frosting separately), but iced orange cookies can also be frozen (they just aren't as easy to stack).
TO FREEZE COOKIE DOUGH: Drop cookie dough balls onto a parchment-lined baking sheet. Freeze until solid. Transfer the frozen cookie dough balls to an airtight freezer-safe storage container or ziptop bag. Freeze for up to 3 months. Bake cookies from frozen as desired (they may need a few extra minutes in the oven).