Preheat oven to 400°F. Line two baking sheets with parchment paper.
Wash, then grate the carrots on the large holes of a box grater (no need to peel first). Add the carrots to a steamer basket and place over simmering water with a lid on top. Steam the carrots for 5-7 minutes to soften. Transfer the softened carrots to a bowl or spread out on a plate to cool.
In a large mixing bowl, beat butter and sugar until creamy and light.
Add the cooled carrots, egg, and vanilla extract. Mix well, stopping to scrape the bottom and sides of the bowl.
Add the flour, baking soda, and salt to the carrot dough. Mix until just combined. Add the nuts at the end, if using.
Make the glaze by combining melted butter, powdered sugar, orange zest, and 2-3 tablespoons of the fresh orange juice in a medium mixing bowl Whisk well until combined, adding more orange juice as needed to get a glaze that is thin enough to dip the cookies in but not too thin. If it is too thin, add more powdered sugar. If too thick, add more orange juice.
Scoop dough onto prepared baking sheet using a small cookie scoop (about 1 heaping tablespoon of dough per cookie). Bake for ten minutes or until the tops are slightly browned. Do not overbake. Cool for 5-10 minutes, then transfer the cookies to a cooling rack set over a baking sheet to get ready for the glaze.
While the cookies are still warm, spoon the glaze onto the cookies or carefully dip the tops of each cookie into the glaze. Let the excess drip off and let the cookies set up for at least 30-60 minutes for the glaze to harden before enjoying.