Preheat oven to 350°F. Line two baking sheets with parchment paper and set aside.
Combine the butter and both sugars in the bowl of a stand mixer. Beat at medium speed until smooth and creamy, about 3 minutes.
Add in the flax egg and vanilla, and beat until combined.
Add the flour, baking soda, cinnamon and salt, and beat until well mixed.
Stir in carrots and pecans.
Using a 2 tbsp cookie scoop, scoop spoonfuls of the cookie dough onto the baking sheets. Bake for about 10-15 minutes or until lightly browned around edges. Cool completely on the sheets.
Meanwhile, make the frosting. Combine the powdered sugar and butter in the clean bowl of a stand mixer and beat until it starts to combine. Add in the the vanilla.
Add in non-dairy milk 1 tbsp at a time until smooth and creamy. You may need to add more or less milk to your liking - you want to have a smooth and spreadable frosting.
Frost the cooled cookies with the icing and eat!