In the mixing bowl, whisk together melted butter, brown sugar, and granulated sugar until smooth.
Beat in egg and vanilla extract until fully incorporated.
Stir in shredded carrots and orange zest.
Add oats, flour, baking soda, cinnamon, nutmeg, and salt; mix until a soft dough forms.
Cover and chill dough for 30 minutes to firm up.
Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
Scoop about 1½ tablespoons of dough per cookie onto prepared sheets, spacing them 2 inches apart.
Bake for 10–12 minutes, or until edges are set and tops are lightly golden.
Let cookies cool on the baking sheets for 5 minutes, then transfer to a cooling rack.
For the drizzle, whisk powdered sugar, cream cheese, milk, and vanilla together until smooth. Drizzle over completely cooled cookies.