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For the carrot cookies: Preheat the oven to 375 degrees F. Line 2 baking sheets with parchment paper.
Cream together the butter and sugar using an electric hand mixer or a stand 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.
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 small 2-ounce ice cream scoop, drop by heaping scoops onto the prepared baking sheets, 15 cookies per sheet. Bake for 8 to10 minutes, rotating halfway through. Transfer to a cooling rack. Frost the cookies with the buttercream frosting while still warm.