Preheat oven to 350°F (175°C). Line a baking sheet with parchment paper.
In a bowl, whisk together flour, baking powder, baking soda, cinnamon, nutmeg, and salt.
In another bowl, beat butter and sugars until light and fluffy. Add egg and vanilla, beat until combined. Mix in carrots.
Gradually add dry ingredients to wet ingredients, mixing until just combined.
Drop tablespoon-sized scoops of batter onto prepared baking sheet. Bake for 10-12 minutes or until edges are golden. Cool completely.
For the filling: Beat cream cheese, butter, and vanilla until smooth. Gradually add powdered sugar, beating until creamy.
Spread filling on the flat side of half the cookies. Top with remaining cookies to form whoopie pies.