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For the buttercream icing: Cream together the butter and powdered sugar in a small bowl until light and fluffy, at least 5 minutes. Mix in the zest and vanilla extract.
For the carrot cookies: Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper.
Cream together the butter and granulated sugar using an electric mixer until light and fluffy, at least 5 minutes. Add in the eggs, one at a time, and beat for 2 minutes after each addition. Mix in the carrot baby food, orange zest and vanilla extract.
Whisk together the flour, baking powder and salt in a separate mixing bowl. Add the dry ingredients to the wet ingredients and beat together on low speed until incorporated, 1 minute.
Using a 2-ounce ice cream scoop, drop the batter by heaping scoops onto the prepared baking sheets, 15 cookies per sheet. Bake for 8 to 10 minutes, rotating halfway through. Transfer to a cooling rack. Frost the cookies with the buttercream icing while still warm.
Baby version: Prepare cookies according to directions above only using a 1-ounce cookie scoop (about 1 tablespoon) and reducing baking time by 2 minutes. Don't frost with buttercream-serve as is.
Special equipment: a 2-ounce ice cream scoop