How to make carrot puree. Peel the carrots and chop them roughly into cubes. Boil the fresh carrots for 20 minutes until very soft. Cool slightly and puree until smooth (with an immersion blender or food processor). Measure 1 cup and wrap it in a linen cloth to remove the excess moisture.
In a bowl of a stand mixer with a paddle attachment or just a large bowl with an electrical mixer, beat the butter and sugar together. Add the egg and vanilla, and mix to combine. Fold in the carrots.
Mix flour, salt, cinnamon, and baking powder in separate bowls. Add the dry ingredients to the wet ingredients. Scrape the sides of the bowl and mix all until combined well, cover with plastic wrap, and refrigerate for 1 hour.
Preheat the oven to 375 degrees Fahrenheit and line a baking sheet with parchment paper.
With a smaller cookie scoop, make 25 cookie dough balls and transfer them to a baking sheet leaving 2 inches apart for cookies to spread. These will spread less, though. Bake for 12-15 minutes. These cookies are slightly crispy on the outside and very soft on the inside.
Let cookies cool completely before the next step.
Make the glaze. Combine the powdered sugar with orange juice and orange zest in a medium bowl and dip each top of the cookie into the glaze one by one. Transfer to a wire rack with a paper towel beneath and allow the glaze to set. Optional: For a pretty presentation, you can add extra orange zest on top before the glaze has set.