Preheat oven to 350°F and line two baking sheets with parchment paper.
Cream butter and granulated sugar on medium speed with a mixer until combined and fluffy. Add vanilla and eggs; mix until just combined.
Whisk together flour, powdered sugar, baking powder, cinnamon, nutmeg, ginger, and salt. Add half of flour mixture to the butter mixture. Mix on low speed until just combined, and then medium speed until completely combined. Repeat with second half of flour mixture. Add grated carrot and pecans and mix until they’re evenly distributed throughout the dough.
Scoop dough onto the prepared baking sheet, arranging about 2-inches apart. (I got 7 cookies on each baking sheet.)
Bake cookies for 7 to 8 minutes, or until crisp around the edges and set in the middle. Let cookies cool on pans for a few minutes before transferring to a cooling rack.
To make the frosting, beat cream cheese and vanilla in the bowl of a stand mixer until smooth. Add half of the powdered sugar and mix on low until mostly combined, then medium-high speed until completely combined. Repeat with second half of powdered sugar.
Spread frosting on cooled cookies and sprinkle with cinnamon and chopped pecans for garnish.